MathsGee Homework Help Q&A - Recent questions and answers in Mathematics
https://mathsgee.com/qa/mathematics
Powered by Question2AnswerAnswered: How are exponential functions related to geometric sequences?
https://mathsgee.com/36831/how-are-exponential-functions-related-geometric-sequences?show=36832#a36832
<p>Geometric sequences are the discrete version of exponential functions, which are continuous.
<br>
</p>
<p><strong>Explanation:</strong>
<br>
Geometric sequences are formed by choosing a starting value and generating each subsequent value by multiplying the previous value by some constant called the geometric ratio. If a formula is provided, terms of the sequence are calculated by substituting \(n=0,1,2,3, \ldots\) into the formula. Note how only whole numbers are used, because it doesn't make sense to have a "one and three-quarterth" term. With an exponential function, the inputs can be any real number from negative infinity to positive infinity.</p>Mathematicshttps://mathsgee.com/36831/how-are-exponential-functions-related-geometric-sequences?show=36832#a36832Thu, 27 Jan 2022 02:25:38 +0000Answered: 3;3;6;9;15;... are the first five terms of a quadratic pattern. Write down the value of the sixth term (T6) of the pattern.
https://mathsgee.com/36810/first-five-terms-quadratic-pattern-write-value-sixth-pattern?show=36830#a36830
\[<br />
a_{n}=3 F_{n}=\frac{3\left(\phi^{n}-(-\phi)^{-n}\right)}{\sqrt{5}}<br />
\]<br />
Explanation:<br />
This is 3 times the standard Fibonacci sequence.<br />
Each term is the sum of the two previous terms, but starting with 3,3 , instead of 1,1 .<br />
The standard Fibonnaci sequence starts:<br />
\[<br />
1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987, \ldots<br />
\]<br />
The terms of the Fibonacci sequence can be defined iteratively as:<br />
\[<br />
\begin{aligned}<br />
&F_{1}=1 \\<br />
&F_{2}=1 \\<br />
&F_{n+2}=F_{n}+F_{n+1}<br />
\end{aligned}<br />
\]<br />
The general term can also be expressed by a formula:<br />
\[<br />
F_{n}=\frac{\phi^{n}-(-\phi)^{-n}}{\sqrt{5}}<br />
\]<br />
<br />
where \(\phi=\frac{1}{2}+\frac{\sqrt{5}}{2} \approx 1.618033988\)<br />
So the formula for a term of our example sequence can be written:<br />
\[<br />
a_{n}=3 F_{n}=\frac{3\left(\phi^{n}-(-\phi)^{-n}\right)}{\sqrt{5}}<br />
\]Mathematicshttps://mathsgee.com/36810/first-five-terms-quadratic-pattern-write-value-sixth-pattern?show=36830#a36830Thu, 27 Jan 2022 02:23:10 +0000Answered: 6;6;9;15;... are the first four terms of a quadratic pattern. Write down the value of the fifth term (T5) of the pattern.
https://mathsgee.com/36828/first-terms-quadratic-pattern-write-down-value-fifth-pattern?show=36829#a36829
<p>The sequence \(6;6;9;15;\ldots\) has a constant second difference of 3 thus it is a quadratic sequence.</p>
<p>The first differences are \(0,3,6,\dots\) respectively.</p>
<p>To get the general term of a quadratic sequence we need to use the formula </p>
<p>\[T_n = an^2+bn+c\]</p>
<p>where \(a;b;c\) are obtained from the following simultaneous equations:</p>
<p><img alt="seq2" src="https://mathsgee.com/?qa=blob&qa_blobid=14082258751553217154" style="height:109px; width:531px"></p>
<p>\(2a=3\)</p>
<p>\(\therefore a=\dfrac{3}{2}\)</p>
<p>\(5a + b = 3\)</p>
<p>\(5(\frac{3}{2})+b=3\)</p>
<p>\(b=\dfrac{-9}{2}\)</p>
<p>\(4a+2b+c=6\)</p>
<p>\( 4(\frac{3}{2})+2(\frac{-9}{2})+c=6\)</p>
<p>\(\therefore c=9\)</p>
<p>Substituting \(a,b,c\) in \(T_n=an^2+bn+c\) gives:</p>
<p>\((\frac{3}{2})25+(\frac{-9}{2})5+9\)</p>
<p>\(=\dfrac{75-45+18}{2} = 24\)</p>Mathematicshttps://mathsgee.com/36828/first-terms-quadratic-pattern-write-down-value-fifth-pattern?show=36829#a36829Thu, 27 Jan 2022 02:18:11 +0000John and Julie had a date one Saturday. They agreed to meet outside the cinema at \(8 \mathrm{pm}\). Julie thought that her watch was 5 minutes fast but in actual fact it was 5 minutes slow.
https://mathsgee.com/36827/saturday-agreed-outside-mathrm-thought-minutes-actual-minutes
John and Julie had a date one Saturday. They agreed to meet outside the cinema at \(8 \mathrm{pm}\). Julie thought that her watch was 5 minutes fast but in actual fact it was 5 minutes slow. John thought that his watch was 5 minutes slow but in actual fact it was 5 minutes fast. Julie deliberately turned up 10 minutes late while John decided to turn up 10 minutes early.<br />
Who turned up first and how long had he/she to wait for the other to arrive?Mathematicshttps://mathsgee.com/36827/saturday-agreed-outside-mathrm-thought-minutes-actual-minutesThu, 27 Jan 2022 02:00:35 +0000Answered: The 7th, 8th and 9th terms of a sequence are 61,69 and 77 respectively. Find a formula to describe this sequence.
https://mathsgee.com/36825/terms-sequence-respectively-formula-describe-this-sequence?show=36826#a36826
<p>Looking at the differences,
<br>
<img alt="seq" src="https://mathsgee.com/?qa=blob&qa_blobid=2484621166229639235" style="height:67px; width:271px">
<br>
you can conclude that the sequence must be of the form
<br>
\[
<br>
u_{n}=8 n+c .
<br>
\]
<br>
For \(n=7, u_{7}=56+c\), giving \(c=5\).
<br>
Thus the formula to describe the sequence is
<br>
\[
<br>
u_{n}=8 n+5 .
<br>
\]</p>Mathematicshttps://mathsgee.com/36825/terms-sequence-respectively-formula-describe-this-sequence?show=36826#a36826Thu, 27 Jan 2022 01:59:18 +0000Answered: Find the first 5 terms of the sequence defined by the formula \[ u_{n}=5 n-4 . \]
https://mathsgee.com/36823/find-the-first-terms-of-the-sequence-defined-by-the-formula-u?show=36824#a36824
\[<br />
\begin{aligned}<br />
&u_{1}=5 \times 1-4=1 \\<br />
&u_{2}=5 \times 2-4=6 \\<br />
&u_{3}=5 \times 3-4=11 \\<br />
&u_{4}=5 \times 4-4=16 \\<br />
&u_{5}=5 \times 5-4=21<br />
\end{aligned}<br />
\]<br />
So the sequence is<br />
\[<br />
1,6,11,16,21, \ldots<br />
\]<br />
Here the terms increase by 5 each time and the formula contains a ' \(5 n\) '.<br />
In general, if the terms of a sequence increase by a constant amount, \(d\), each time, then the sequence will be defined by the formula<br />
\[<br />
u_{n}=d n+c<br />
\]<br />
where \(c\) is a constant number.Mathematicshttps://mathsgee.com/36823/find-the-first-terms-of-the-sequence-defined-by-the-formula-u?show=36824#a36824Thu, 27 Jan 2022 01:57:03 +0000Answered: Find the first 5 terms of the sequence defined by the formula \(u_{n}=3 n+6\).
https://mathsgee.com/36821/find-the-first-terms-of-the-sequence-defined-by-the-formula-u?show=36822#a36822
\(u_{1}=3 \times 1+6=9\) \(u_{2}=3 \times 2+6=12\) \(u_{3}=3 \times 3+6=15\) \(u_{4}=3 \times 4+6=18\) \(u_{5}=3 \times 5+6=21\) So the sequence is \(9,12,15,18,21, \ldots\)<br />
Note that the terms of the sequence increase by 3 each time and that the formula contains a ' \(3 n\) '.Mathematicshttps://mathsgee.com/36821/find-the-first-terms-of-the-sequence-defined-by-the-formula-u?show=36822#a36822Thu, 27 Jan 2022 01:55:52 +0000Answered: Find the general term of the sequence \[ 1,4,9,16,25, \ldots \]
https://mathsgee.com/36819/find-the-general-term-of-the-sequence-1-4-9-16-25-ldots?show=36820#a36820
\[<br />
\begin{aligned}<br />
&1,4,9,16,25, \ldots \\<br />
&u_{1}=1=1 \times 1 \\<br />
&u_{2}=4=2 \times 2 \\<br />
&u_{3}=9=3 \times 3 \\<br />
&u_{4}=16=4 \times 4 \\<br />
&u_{5}=25=5 \times 5<br />
\end{aligned}<br />
\]<br />
This sequence can be described by the general formula<br />
\[<br />
u_{n}=n^{2} .<br />
\]Mathematicshttps://mathsgee.com/36819/find-the-general-term-of-the-sequence-1-4-9-16-25-ldots?show=36820#a36820Thu, 27 Jan 2022 01:54:45 +0000Answered: Find the 20th term of the sequence \[ 2,5,10,17,26,37, \ldots \]
https://mathsgee.com/36817/find-the-20th-term-of-the-sequence-2-5-10-17-26-37-ldots?show=36818#a36818
The terms of this sequence can be expressed as<br />
\(\begin{aligned} 1 \text { st term } &=1^{2}+1 \\ 2 \text { nd term } &=2^{2}+1 \\ 3 \text { rd term } &=3^{2}+1 \\ 4 \text { th term } &=4^{2}+1 \\ 5 \text { th term } &=5^{2}+1 \end{aligned}\)<br />
Extending the pattern gives<br />
20 th term \(=20^{2}+1=401\).Mathematicshttps://mathsgee.com/36817/find-the-20th-term-of-the-sequence-2-5-10-17-26-37-ldots?show=36818#a36818Thu, 27 Jan 2022 01:53:00 +0000Answered: Find the 10th and 100th terms of the sequence \(3,5,7,9,11, \ldots\)
https://mathsgee.com/36815/find-the-10th-and-100th-terms-of-the-sequence-3-5-7-9-11-ldots?show=36816#a36816
The terms above are given by<br />
\(\begin{aligned} \text { 1st term } &=3 \\ \text { 2nd term } &=3+2=5 \\ \text { 3rd term } &=3+2 \times 2=7 \\ \text { 4th term } &=3+3 \times 2=9 \\ \text { 5th term } &=3+4 \times 2=11 \end{aligned}\)<br />
This can be extended to give<br />
\[<br />
\begin{aligned}<br />
10 \text { th term } &=3+9 \times 2=21 \\<br />
100 \text { th term } &=3+99 \times 2=201<br />
\end{aligned}<br />
\]Mathematicshttps://mathsgee.com/36815/find-the-10th-and-100th-terms-of-the-sequence-3-5-7-9-11-ldots?show=36816#a36816Thu, 27 Jan 2022 01:52:00 +0000Answered: Find the 20th term of the sequence \(8,16,24,32, \ldots\)
https://mathsgee.com/36813/find-the-20th-term-of-the-sequence-8-16-24-32-ldots?show=36814#a36814
The terms of the sequence can be obtained as shown below.<br />
\(\begin{array}{ll}\text { 1st term } & =1 \times 8=8 \\ \text { 2nd term } & =2 \times 8=16 \\ \text { 3rd term } & =3 \times 8=24 \\ \text { 4th term } & =4 \times 8=32\end{array}\)<br />
This pattern can be extended to give<br />
20 th term \(=20 \times 8=160\)Mathematicshttps://mathsgee.com/36813/find-the-20th-term-of-the-sequence-8-16-24-32-ldots?show=36814#a36814Thu, 27 Jan 2022 01:50:48 +0000Answered: What is the derivative of $y = x^2$?
https://mathsgee.com/36747/what-is-the-derivative-of-y-x-2?show=36776#a36776
If \(y=ax^n\) then \(\dfrac{dy}{dx}=a\cdot n \cdot x^{n-1}\)<br />
<br />
Since \(y=x^2\) then \(\dfrac{dy}{dx}=2 \cdot x^{2-1} =2 \cdot x^{1} = 2x\)Mathematicshttps://mathsgee.com/36747/what-is-the-derivative-of-y-x-2?show=36776#a36776Wed, 26 Jan 2022 01:44:35 +0000What is an n-tuple of real numbers?
https://mathsgee.com/36775/what-is-an-n-tuple-of-real-numbers
What is an n-tuple of real numbers?Mathematicshttps://mathsgee.com/36775/what-is-an-n-tuple-of-real-numbersWed, 26 Jan 2022 01:35:25 +0000Answered: How do engineers and physicists represent 2-D and 3-D vectors?
https://mathsgee.com/36773/how-do-engineers-and-physicists-represent-2-d-and-3-d-vectors?show=36774#a36774
<p>Engineers and physicists represent vectors in two dimensions (also called 2-space) or in three dimensions (also called 3-space) by arrows.</p>
<p>The direction of the arrowhead specifies the direction of the vector and the length of the arrow specifies the magnitude. Mathematicians call these geometric vectors. The tail of the arrow is called the initial point of the vector and the tip the terminal point.</p>
<p><img alt="terminal" src="https://mathsgee.com/?qa=blob&qa_blobid=13753159281312841139" style="height:135px; width:256px"></p>Mathematicshttps://mathsgee.com/36773/how-do-engineers-and-physicists-represent-2-d-and-3-d-vectors?show=36774#a36774Wed, 26 Jan 2022 01:34:26 +0000Answered: How do math textbooks denote vector and scalar quantities?
https://mathsgee.com/36771/how-do-math-textbooks-denote-vector-and-scalar-quantities?show=36772#a36772
Vectors are denoted in boldface type such as $\mathbf{a}, \mathbf{b}, \mathbf{v}, \mathbf{w}$, and $\mathbf{x}$, and scalars are denoted in lowercase italic type such as $a, k, v, w$, and $x$. To indicate that a vector $\mathbf{v}$ has initial point $A$ and terminal point $B$, then: $$ \mathbf{v}=\overrightarrow{A B} $$Mathematicshttps://mathsgee.com/36771/how-do-math-textbooks-denote-vector-and-scalar-quantities?show=36772#a36772Wed, 26 Jan 2022 01:32:48 +0000Answered: When are two vectors considered to be equivalent?
https://mathsgee.com/36769/when-are-two-vectors-considered-to-be-equivalent?show=36770#a36770
<p>Vectors with the same length and direction, such as those below, are said to be equivalent.</p>
<p>Since we want a vector to be determined solely by its length and direction, equivalent vectors are regarded as the same vector even though they may be in different positions. Equivalent vectors are also said to be equal, which we indicate by writing
<br>
$$
<br>
\mathbf{v}=\mathbf{w}
<br>
$$</p>
<p><img alt="ersdaas" src="https://mathsgee.com/?qa=blob&qa_blobid=13665544512842133195" style="height:118px; width:231px"></p>Mathematicshttps://mathsgee.com/36769/when-are-two-vectors-considered-to-be-equivalent?show=36770#a36770Wed, 26 Jan 2022 01:30:56 +0000What is the difference between vectors and matrices?
https://mathsgee.com/36768/what-is-the-difference-between-vectors-and-matrices
What is the difference between vectors and matrices?Mathematicshttps://mathsgee.com/36768/what-is-the-difference-between-vectors-and-matricesWed, 26 Jan 2022 01:29:22 +0000Answered: What is a zero vector?
https://mathsgee.com/36766/what-is-a-zero-vector?show=36767#a36767
The vector whose initial and terminal points coincide has length zero, so we call this the zero vector and denote it by $\mathbf{0}$. The zero vector has no natural direction, so we will agree that it can be assigned any direction that is convenient for the problem at hand.Mathematicshttps://mathsgee.com/36766/what-is-a-zero-vector?show=36767#a36767Wed, 26 Jan 2022 01:28:07 +0000Answered: How do you define the Parallelogram Rule of Vector Addition?
https://mathsgee.com/36764/how-do-you-define-the-parallelogram-rule-of-vector-addition?show=36765#a36765
<p><strong>Parallelogram Rule for Vector Addition</strong></p>
<p>If $\mathbf{v}$ and $\mathbf{w}$ are vectors in 2 -space or 3 -space that are positioned so their initial points coincide, then the two vectors form adjacent sides of a parallelogram, and the $\operatorname{sum} \mathbf{v}+\mathbf{w}$ is the vector represented by the arrow from the common initial point of $\mathbf{v}$ and $\mathbf{w}$ to the opposite vertex of the parallelogram.</p>
<p><img alt="rect" src="https://mathsgee.com/?qa=blob&qa_blobid=12885900816632513174" style="height:135px; width:195px"></p>Mathematicshttps://mathsgee.com/36764/how-do-you-define-the-parallelogram-rule-of-vector-addition?show=36765#a36765Wed, 26 Jan 2022 01:26:51 +0000Answered: What does the Triangle Rule of Vector Addition state?
https://mathsgee.com/36762/what-does-the-triangle-rule-of-vector-addition-state?show=36763#a36763
<p><strong>Triangle Rule for Vector Addition</strong></p>
<p>If $\mathbf{v}$ and $\mathbf{w}$ are vectors in 2 -space or 3 -space that are positioned so the initial point of $\mathbf{w}$ is at the terminal point of $\mathbf{v}$, then the $\boldsymbol{s u m} \mathbf{v}+\mathbf{w}$ is represented by the arrow from the initial point of $\mathbf{v}$ to the terminal point of $\mathbf{w}$</p>Mathematicshttps://mathsgee.com/36762/what-does-the-triangle-rule-of-vector-addition-state?show=36763#a36763Wed, 26 Jan 2022 01:25:10 +0000Answered: What are the objects that linear algebra is mainly concerned about?
https://mathsgee.com/36760/what-are-objects-that-linear-algebra-mainly-concerned-about?show=36761#a36761
Linear algebra is primarily concerned with two types of mathematical objects, "matrices" and "vectors."Mathematicshttps://mathsgee.com/36760/what-are-objects-that-linear-algebra-mainly-concerned-about?show=36761#a36761Wed, 26 Jan 2022 01:22:44 +0000Answered: What kind of a transformation is vector addition?
https://mathsgee.com/36758/what-kind-of-a-transformation-is-vector-addition?show=36759#a36759
Vector addition can also be viewed as a process of translating points.Mathematicshttps://mathsgee.com/36758/what-kind-of-a-transformation-is-vector-addition?show=36759#a36759Wed, 26 Jan 2022 01:21:32 +0000Answered: Does the arithmetic rule $a-b = a + (-b)$, apply for vector arithmetic?
https://mathsgee.com/36756/does-the-arithmetic-rule-a-b-a-b-apply-for-vector-arithmetic?show=36757#a36757
In ordinary arithmetic we can write $a-b=a+(-b)$, which expresses subtraction in terms of addition. There is an analogous idea in vector arithmetic.Mathematicshttps://mathsgee.com/36756/does-the-arithmetic-rule-a-b-a-b-apply-for-vector-arithmetic?show=36757#a36757Wed, 26 Jan 2022 01:20:15 +0000Answered: What is the definition for Vector Subtraction?
https://mathsgee.com/36754/what-is-the-definition-for-vector-subtraction?show=36755#a36755
<p><strong>Vector Subtraction</strong></p>
<p>The negative of a vector $\mathbf{v}$, denoted by $-\mathbf{v}$, is the vector that has the same length as $\mathbf{v}$ but is oppositely directed, and the difference of $\mathbf{v}$ from $\mathbf{w}$, denoted by $\mathbf{w}-\mathbf{v}$, is taken to be the sum
<br>
$$
<br>
\mathbf{w}-\mathbf{v}=\mathbf{w}+(-\mathbf{v})
<br>
$$</p>Mathematicshttps://mathsgee.com/36754/what-is-the-definition-for-vector-subtraction?show=36755#a36755Wed, 26 Jan 2022 01:18:58 +0000Answered: How do I represent the difference between two vectors geometrically?
https://mathsgee.com/36752/how-represent-the-difference-between-vectors-geometrically?show=36753#a36753
<p>The difference of $\mathbf{v}$ from $\mathbf{w}$ can be obtained geometrically by the parallelogram method as shown in the diagram below.</p>
<p><img alt="erdfe" src="https://mathsgee.com/?qa=blob&qa_blobid=10499187219881652126" style="height:117px; width:517px"></p>Mathematicshttps://mathsgee.com/36752/how-represent-the-difference-between-vectors-geometrically?show=36753#a36753Wed, 26 Jan 2022 01:17:45 +0000Answered: When dealing with vectors, what does scalar multiplication mean?
https://mathsgee.com/36750/when-dealing-with-vectors-what-does-scalar-multiplication?show=36751#a36751
<p>Scalar Multiplication If $\mathbf{v}$ is a nonzero vector in 2 -space or 3 -space, and if $k$ is a nonzero scalar, then we define the scalar product of $\mathbf{v}$ by $\mathbf{k}$ to be the vector whose length is $|k|$ times the length of $\mathbf{v}$ and whose direction is the same as that of $\mathbf{v}$ if $k$ is positive and opposite to that of $\mathbf{v}$ if $k$ is negative. If $k=0$ or $\mathbf{v}=\mathbf{0}$, then we define $k \mathbf{v}$ to be $\mathbf{0}$.</p>
<p>The diagram below shows the geometric relationship between a vector <strong>v</strong> and some of its scalar multiples. In particular, observe that <strong>(-1)v</strong> has the same length as $\mathbf{v}$ but is oppositely directed; therefore,
<br>
$$
<br>
(-1) \mathbf{v}=-\mathbf{v}
<br>
$$</p>
<p><img alt="lines" src="https://mathsgee.com/?qa=blob&qa_blobid=12239046056674017258" style="height:214px; width:267px"></p>Mathematicshttps://mathsgee.com/36750/when-dealing-with-vectors-what-does-scalar-multiplication?show=36751#a36751Wed, 26 Jan 2022 01:15:51 +0000Answered: What are parallel and collinear vectors?
https://mathsgee.com/36748/what-are-parallel-and-collinear-vectors?show=36749#a36749
<p>Suppose that $\mathbf{v}$ and $\mathbf{w}$ are vectors in 2 -space or 3 -space with a common initial point. If one of the vectors is a scalar multiple of the other, then the vectors lie on a common line, so it is reasonable to say that they are collinear.</p>
<p>However, if we translate one of the vectors, then the vectors are parallel but no longer collinear. This creates a linguistic problem because translating a vector does not change it.</p>
<p>The only way to resolve this problem is to agree that the terms parallel and collinear mean the same thing when applied to vectors. Although the vector $\mathbf{0}$ has no clearly defined direction, we will regard it as parallel to all vectors when convenient.</p>
<p><img alt="parallel" src="https://mathsgee.com/?qa=blob&qa_blobid=5248816641511352955" style="height:171px; width:559px"></p>Mathematicshttps://mathsgee.com/36748/what-are-parallel-and-collinear-vectors?show=36749#a36749Wed, 26 Jan 2022 01:13:52 +0000Answered: Does vector addition satisfy the associative law of addition?
https://mathsgee.com/36745/does-vector-addition-satisfy-the-associative-law-addition?show=36746#a36746
Vector addition satisfies the associative law for addition, meaning that when we add three vectors, say $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$, it does not matter which two we add first; that is, $$ \mathbf{u}+(\mathbf{v}+\mathbf{w})=(\mathbf{u}+\mathbf{v})+\mathbf{w} $$ It follows from this that there is no ambiguity in the expression $\mathbf{u}+\mathbf{v}+\mathbf{w}$ because the same result is obtained no matter how the vectors are grouped.Mathematicshttps://mathsgee.com/36745/does-vector-addition-satisfy-the-associative-law-addition?show=36746#a36746Wed, 26 Jan 2022 01:11:24 +0000Answered: What are the components of a zero vector?
https://mathsgee.com/36743/what-are-the-components-of-a-zero-vector?show=36744#a36744
The component forms of the zero vector are $\mathbf{0}=(0,0)$ in 2-space and $\mathbf{0}=(0,0,0)$ in 3-space.Mathematicshttps://mathsgee.com/36743/what-are-the-components-of-a-zero-vector?show=36744#a36744Wed, 26 Jan 2022 01:10:07 +0000Answered: What are components of a vector?
https://mathsgee.com/36741/what-are-components-of-a-vector?show=36742#a36742
If a vector $\mathbf{v}$ in 2 -space or 3 -space is positioned with its initial point at the origin of a rectangular coordinate system, then the vector is completely determined by the coordinates of its terminal point. We call these coordinates the components of $\mathbf{v}$ relative to the coordinate system. We will write $\mathbf{v}=\left(v_{1}, v_{2}\right)$ to denote a vector $\mathbf{v}$ in 2-space with components $\left(v_{1}, v_{2}\right)$, and $\mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right)$ to denote a vector $\mathbf{v}$ in 3 -space with components $\left(v_{1}, v_{2}, v_{3}\right)$.Mathematicshttps://mathsgee.com/36741/what-are-components-of-a-vector?show=36742#a36742Wed, 26 Jan 2022 01:09:01 +0000Answered: What does vector multiplication achieve?
https://mathsgee.com/36739/what-does-vector-multiplication-achieve?show=36740#a36740
Sometimes there is a need to change the length of a vector or change its length and reverse its direction. This is accomplished by a type of multiplication in which vectors are multiplied by scalars. As an example, the product $2 \mathrm{v}$ denotes the vector that has the same direction as $\mathbf{v}$ but twice the length, and the product $-2 \mathbf{v}$ denotes the vector that is oppositely directed to $\mathbf{v}$ and has twice the length.Mathematicshttps://mathsgee.com/36739/what-does-vector-multiplication-achieve?show=36740#a36740Wed, 26 Jan 2022 01:07:54 +0000Answered: What is scalar multiplication of vectors?
https://mathsgee.com/36737/what-is-scalar-multiplication-of-vectors?show=36738#a36738
Scalar Multiplication If $\mathbf{v}$ is a nonzero vector in 2 -space or 3 -space, and if $k$ is a nonzero scalar, then we define the scalar product of $\mathbf{v}$ by $\mathrm{k}$ to be the vector whose length is $|k|$ times the length of $\mathbf{v}$ and whose direction is the same as that of $\mathbf{v}$ if $k$ is positive and opposite to that of $\mathbf{v}$ if $k$ is negative. If $k=0$ or $\mathbf{v}=\mathbf{0}$, then we define $k \mathbf{v}$ to be $\mathbf{0}$.Mathematicshttps://mathsgee.com/36737/what-is-scalar-multiplication-of-vectors?show=36738#a36738Wed, 26 Jan 2022 01:06:45 +0000Answered: What are collinear vectors?
https://mathsgee.com/36735/what-are-collinear-vectors?show=36736#a36736
Suppose that $\mathbf{v}$ and $\mathbf{w}$ are vectors in 2 -space or 3 -space with a common initial point. If one of the vectors is a scalar multiple of the other, then the vectors lie on a common line, so it is reasonable to say that they are collinear.Mathematicshttps://mathsgee.com/36735/what-are-collinear-vectors?show=36736#a36736Wed, 26 Jan 2022 01:05:30 +0000Answered: What is the difference between collinear vectors and parallel vectors?
https://mathsgee.com/36733/what-difference-between-collinear-vectors-parallel-vectors?show=36734#a36734
<p>Suppose that $\mathbf{v}$ and $\mathbf{w}$ are vectors in 2 -space or 3 -space with a common initial point. If one of the vectors is a scalar multiple of the other, then the vectors lie on a common line, so it is reasonable to say that they are collinear.</p>
<p>However, if we translate one of the vectors, then the vectors are <strong>parallel </strong>but no longer <strong>collinear</strong>.</p>
<p>This creates a linguistic problem because translating a vector does not change it. The only way to resolve this problem is to agree that the terms parallel and collinear mean the same thing when applied to vectors.</p>
<p>Although the vector $\mathbf{0}$ has no clearly defined direction, we will regard it as parallel to all vectors when convenient.</p>Mathematicshttps://mathsgee.com/36733/what-difference-between-collinear-vectors-parallel-vectors?show=36734#a36734Wed, 26 Jan 2022 01:04:19 +0000Answered: How can you geometrically show that two vectors in 2-space or 3-space are equivalent?
https://mathsgee.com/36731/how-geometrically-show-that-vectors-space-space-equivalent?show=36732#a36732
If a vector $\mathbf{v}$ in 2 -space or 3 -space is positioned with its initial point at the origin of a rectangular coordinate system, then the vector is completely determined by the coordinates of its terminal point. We call these coordinates the components of $\mathbf{v}$ relative to the coordinate system. We will write $\mathbf{v}=\left(v_{1}, v_{2}\right)$ to denote a vector $\mathbf{v}$ in 2 -space with components $\left(v_{1}, v_{2}\right)$, and $\mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right)$ to denote a vector $\mathbf{v}$ in 3 -space with components $\left(v_{1}, v_{2}, v_{3}\right)$. It should be evident geometrically that two vectors in 2 -space or 3 -space are equivalent if and only if they have the same terminal point when their initial points are at the origin. Algebraically, this means that two vectors are equivalent if and only if their corresponding components are equal. Thus, for example, the vectors $$ \mathbf{v}=\left(v_{1}, v_{2}, v_{3}\right) \quad \text { and } \mathbf{w}=\left(w_{1}, w_{2}, w_{3}\right) $$ in 3 -space are equivalent if and only if $$ v_{1}=w_{1}, \quad v_{2}=w_{2}, \quad v_{3}=w_{3} $$Mathematicshttps://mathsgee.com/36731/how-geometrically-show-that-vectors-space-space-equivalent?show=36732#a36732Wed, 26 Jan 2022 01:02:58 +0000Answered: How do I write vectors whose initial point is not at the origin?
https://mathsgee.com/36729/how-do-write-vectors-whose-initial-point-is-not-at-the-origin?show=36730#a36730
<p>It is sometimes necessary to consider vectors whose initial points are not at the origin. If $\overrightarrow{P_{1} P_{2}}$ denotes the vector with initial point $P_{1}\left(x_{1}, y_{1}\right)$ and terminal point $P_{2}\left(x_{2}, y_{2}\right)$, then the components of this vector are given by the formula
<br>
$$
<br>
\overrightarrow{P_{1} P_{2}}=\left(x_{2}-x_{1}, y_{2}-y_{1}\right)
<br>
$$
<br>
That is, the components of $\overrightarrow{P_{1} P_{2}}$ are obtained by subtracting the coordinates of the initial point from the coordinates of the terminal point. For example, in Figure 3.1.12 the vector $\overrightarrow{P_{1} P_{2}}$ is the difference of vectors $\overrightarrow{O P_{2}}$ and $\overrightarrow{O P_{1}}$, so
<br>
$$
<br>
\overrightarrow{P_{1} P_{2}}=\overrightarrow{O P_{2}}-\overrightarrow{O P_{1}}=\left(x_{2}, y_{2}\right)-\left(x_{1}, y_{1}\right)=\left(x_{2}-x_{1}, y_{2}-y_{1}\right)
<br>
$$
<br>
As you might expect, the components of a vector in 3 -space that has initial point $P_{1}\left(x_{1}, y_{1}, z_{1}\right)$ and terminal point $P_{2}\left(x_{2}, y_{2}, z_{2}\right)$ are given by
<br>
$$
<br>
\overrightarrow{P_{1} P_{2}}=\left(x_{2}-x_{1}, y_{2}-y_{1}, z_{2}-z_{1}\right)
<br>
$$</p>
<p> </p>
<p><img alt="triangle2" src="https://mathsgee.com/?qa=blob&qa_blobid=2761104111805898059" style="height:163px; width:222px"></p>
<p> </p>Mathematicshttps://mathsgee.com/36729/how-do-write-vectors-whose-initial-point-is-not-at-the-origin?show=36730#a36730Wed, 26 Jan 2022 00:58:27 +0000Answered: The components of the vector $\mathbf{v}=\overrightarrow{P_{1} P_{2}}$ with initial point $P_{1}(2,-1,4)$ and terminal point $P_{2}(7,5,-8)$ are
https://mathsgee.com/36727/components-vector-mathbf-overrightarrow-initial-terminal?show=36728#a36728
<p>Using the formula:</p>
<p>$$
<br>
\overrightarrow{P_{1} P_{2}}=\left(x_{2}-x_{1}, y_{2}-y_{1}, z_{2}-z_{1}\right)
<br>
$$</p>
<p>we can determine the components of <strong>v</strong></p>
<p><strong>$$
<br>
\mathbf{v}=(7-2,5-(-1),(-8)-4)=(5,6,-12)
<br>
$$</strong></p>Mathematicshttps://mathsgee.com/36727/components-vector-mathbf-overrightarrow-initial-terminal?show=36728#a36728Wed, 26 Jan 2022 00:55:56 +0000Answered: What is an n-space in algebra?
https://mathsgee.com/36725/what-is-an-n-space-in-algebra?show=36726#a36726
The $n$ in $n$-space represents a natural number i.e. $n \in \mathbb{N}$ The $n$-space is a higher dimesional space of order $n$. The idea of using ordered pairs and triples of real numbers to represent points in twodimensional space and three-dimensional space was well known in the eighteenth and nineteenth centuries. By the dawn of the twentieth century, mathematicians and physicists were exploring the use of "higher dimensional" spaces in mathematics and physics. Today, even the layman is familiar with the notion of time as a fourth dimension, an idea used by Albert Einstein in developing the general theory of relativity. Today, physicists working in the field of "string theory" commonly use 11 -dimensional space in their quest for a unified theory that will explain how the fundamental forces of nature work. Much of the remaining work in this section is concerned with extending the notion of space to $n$ dimensions.Mathematicshttps://mathsgee.com/36725/what-is-an-n-space-in-algebra?show=36726#a36726Wed, 26 Jan 2022 00:54:34 +0000Answered: What are n-tuples?
https://mathsgee.com/36723/what-are-n-tuples?show=36724#a36724
To explore these ideas further, we start with some terminology and notation. The set of all real numbers can be viewed geometrically as a line. It is called the real line and is denoted by $R$ or $R^{1}$. The superscript reinforces the intuitive idea that a line is onedimensional. The set of all ordered pairs of real numbers (called 2-tuples) and the set of all ordered triples of real numbers (called 3-tuples) are denoted by $R^{2}$ and $R^{3}$, respectively. The superscript reinforces the idea that the ordered pairs correspond to points in the plane (two-dimensional) and ordered triples to points in space (three-dimensional). The following definition extends this idea.Mathematicshttps://mathsgee.com/36723/what-are-n-tuples?show=36724#a36724Wed, 26 Jan 2022 00:53:15 +0000Answered: What is the definition of an ordered n-tuple?
https://mathsgee.com/36721/what-is-the-definition-of-an-ordered-n-tuple?show=36722#a36722
If $n$ is a positive integer, then an ordered $n$-tuple is a sequence of $n$ real numbers $\left(v_{1}, v_{2}, \ldots, v_{n}\right) .$ The set of all ordered $n$-tuples is called $\boldsymbol{n}$-space and is denoted by $R^{n}$. You can think of the numbers in an $n$-tuple $\left(v_{1}, v_{2}, \ldots, v_{n}\right)$ as either the coordinates of a generalized point or the components of a generalized vector, depending on the geometric image you want to bring to mind - the choice makes no difference mathematically, since it is the algebraic properties of $n$-tuples that are of concern.Mathematicshttps://mathsgee.com/36721/what-is-the-definition-of-an-ordered-n-tuple?show=36722#a36722Wed, 26 Jan 2022 00:52:05 +0000Answered: What are some typical applications that lead to n-tuples?
https://mathsgee.com/36719/what-are-some-typical-applications-that-lead-to-n-tuples?show=36720#a36720
<p>Here are some typical applications that lead to n-tuples.</p>
<ul>
<li><strong>Experimental Data </strong>- A scientist performs an experiment and makes $n$ numerical measurements each time the experiment is performed. The result of each experiment can be regarded as a vector $\mathbf{y}=\left(y_{1}, y_{2}, \ldots, y_{n}\right)$ in $R^{n}$ in which $y_{1}, y_{2}, \ldots, y_{n}$ are the measured values.</li>
<li><strong>Storage and Warehousing</strong> - A national trucking company has 15 depots for storing and servicing its trucks. At each point in time the distribution of trucks in the service depots can be described by a 15-tuple $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{15}\right)$ in which $x_{1}$ is the number of trucks in the first depot, $x_{2}$ is the number in the second depot, and so forth.</li>
<li><strong>Electrical Circuits </strong>- A certain kind of processing chip is designed to receive four input voltages and produce three output voltages in response. The input voltages can bé régarded as vectors in $R^{4}$ and tháº» output voltagess *** vectors in $R^{3}$. Thus, thè chip can be viewed as a device that transforms an input vector $\mathbf{v}=\left(v_{1}, v_{2}, v_{3}, v_{4}\right)$ in $R^{4}$ into an output vector $\mathbf{w}=\left(w_{1}, w_{2}, w_{3}\right)$ in $R^{3}$.</li>
<li><strong>Graphical Images</strong> - One way in which color images are created on computer screens is by assigning each pixel (an addressable point on the screen) three numbers that describe the hue, saturation, and brightness of the pixel. Thus, a complete color image can be viewed as a set of 5 -tuples of the form $\mathbf{v}=(x, y, h, s, b)$ in which $x$ and $y$ are the screen coordinates of a pixel and $h, s$, and $b$ are its hue, saturation, and brightness.</li>
<li><strong>Economics</strong> - One approach to economic analysis is to divide an economy into sectors (manufacturing, services, utilities, and so forth) and measure the output of each sector by a dollar value. Thus, in an economy with 10 sectors the economic output of the entire economy can be represented by a 10-tuple $\mathbf{s}=\left(s_{1}, s_{2}, \ldots, s_{10}\right)$ in which the numbers $s_{1}, s_{2}, \ldots, s_{10}$ are the outputs of the individual sectors.</li>
<li><strong>Mechanical Systems</strong> - Suppose that six particles move along the same coordinate line so that at time $t$ their coordinates are $x_{1}, x_{2}, \ldots, x_{6}$ and their velocities are $v_{1}, v_{2}, \ldots, v_{6}$, respectively. This information can be represented by the vector
<br>
$$
<br>
\mathbf{v}=\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, v_{1}, v_{2}, v_{3}, v_{4}, v_{5}, v_{6}, t\right)
<br>
$$
<br>
in $R^{13} .$ This vector is called the state of the particle system at time $t$.</li>
</ul>Mathematicshttps://mathsgee.com/36719/what-are-some-typical-applications-that-lead-to-n-tuples?show=36720#a36720Wed, 26 Jan 2022 00:50:45 +0000Answered: When are vectors in $\mathbb{R}^n$ equivalent?
https://mathsgee.com/36717/when-are-vectors-in-mathbb-r-n-equivalent?show=36718#a36718
Vectors $\mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right)$ and $\mathbf{w}=\left(w_{1}, w_{2}, \ldots, w_{n}\right)$ in $R^{n}$ are said to be equivalent (also called equal) if $$ v_{1}=w_{1}, \quad v_{2}=w_{2}, \ldots, \quad v_{n}=w_{n} $$ We indicate this by writing $\mathbf{v}=\mathbf{w}$.Mathematicshttps://mathsgee.com/36717/when-are-vectors-in-mathbb-r-n-equivalent?show=36718#a36718Wed, 26 Jan 2022 00:49:27 +0000Answered: When are the two vectors $(a,b,c,d)$ and $(1,-4,2,7)$ equivalent?
https://mathsgee.com/36715/when-are-the-two-vectors-a-b-c-d-and-1-4-2-7-equivalent?show=36716#a36716
$(a, b, c, d)=(1,-4,2,7)$ if and only if $a=1, b=-4, c=2$, and $d=7$Mathematicshttps://mathsgee.com/36715/when-are-the-two-vectors-a-b-c-d-and-1-4-2-7-equivalent?show=36716#a36716Wed, 26 Jan 2022 00:48:20 +0000Answered: How are the operations of addition, subtraction and scalar multiplication defined for vectors in $\mathbb{R}^n$?
https://mathsgee.com/36713/operations-addition-subtraction-multiplication-defined?show=36714#a36714
If $\mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right)$ and $\mathbf{w}=\left(w_{1}, w_{2}, \ldots, w_{n}\right)$ are vectors in $R^{n}$, and if $k$ is any scalar, then we define $$ \begin{aligned} &\mathbf{v}+\mathbf{w}=\left(v_{1}+w_{1}, v_{2}+w_{2}, \ldots, v_{n}+w_{n}\right) \\ &k \mathbf{v}=\left(k v_{1}, k v_{2}, \ldots, k v_{n}\right) \\ &-\mathbf{v}=\left(-v_{1},-v_{2}, \ldots,-v_{n}\right) \\ &\mathbf{w}-\mathbf{v}=\mathbf{w}+(-\mathbf{v})=\left(w_{1}-v_{1}, w_{2}-v_{2}, \ldots, w_{n}-v_{n}\right) \end{aligned} $$Mathematicshttps://mathsgee.com/36713/operations-addition-subtraction-multiplication-defined?show=36714#a36714Wed, 26 Jan 2022 00:46:55 +0000Answered: If $\mathbf{v}=(1,-3,2)$ and $\mathbf{w}=(4,2,1)$, then
https://mathsgee.com/36711/if-mathbf-v-1-3-2-and-mathbf-w-4-2-1-then?show=36712#a36712
If $\mathbf{v}=(1,-3,2)$ and $\mathbf{w}=(4,2,1)$, then $$ \begin{array}{ll} \mathbf{v}+\mathbf{w}=(5,-1,3), & 2 \mathbf{v}=(2,-6,4) \\ -\mathbf{w}=(-4,-2,-1), & \mathbf{v}-\mathbf{w}=\mathbf{v}+(-\mathbf{w})=(-3,-5,1) \end{array} $$Mathematicshttps://mathsgee.com/36711/if-mathbf-v-1-3-2-and-mathbf-w-4-2-1-then?show=36712#a36712Wed, 26 Jan 2022 00:45:15 +0000Prove that if $\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ are vectors in $R^{n}$, and if $k$ and $m$ are scalars, then:
https://mathsgee.com/36710/prove-that-mathbf-mathbf-mathbf-are-vectors-and-scalars-then
Prove that if<br />
<br />
$\mathbf{u}, \mathbf{v}$, and $\mathbf{w}$ are vectors in $R^{n}$, and if $k$ and $m$ are scalars, then: (a) $ \mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$ (b) $ (\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$ (c) $ \mathbf{u}+\mathbf{0}=\mathbf{0}+\mathbf{u}=\mathbf{u}$ (d) $\mathbf{u}+(-\mathbf{u})=\mathbf{0}$ (e) $k(\mathbf{u}+\mathbf{v})=k \mathbf{u}+k \mathbf{v}$ (f) $(k+m) \mathbf{u}=k \mathbf{u}+m \mathbf{u}$ (g) $k(m \mathbf{u})=(k m) \mathbf{u}$ (h) $ \mathbf{l u}=\mathbf{u}$Mathematicshttps://mathsgee.com/36710/prove-that-mathbf-mathbf-mathbf-are-vectors-and-scalars-thenWed, 26 Jan 2022 00:44:06 +0000Prove that If $\mathbf{v}$ is a vector in $R^{n}$ and $k$ is a scalar, then:
https://mathsgee.com/36709/prove-that-if-mathbf-v-is-a-vector-in-r-and-k-is-scalar-then
Prove that If $\mathbf{v}$ is a vector in $R^{n}$ and $k$ is a scalar, then: (a) $0 \mathbf{v}=\mathbf{0}$ (b) $k \mathbf{0}=\mathbf{0}$ (c) $\quad(-1) \mathbf{v}=-\mathbf{v}$Mathematicshttps://mathsgee.com/36709/prove-that-if-mathbf-v-is-a-vector-in-r-and-k-is-scalar-thenWed, 26 Jan 2022 00:43:17 +0000Answered: What is a linear combination of vectors?
https://mathsgee.com/36707/what-is-a-linear-combination-of-vectors?show=36708#a36708
If $\mathbf{w}$ is a vector in $R^{n}$, then $\mathbf{w}$ is said to be a linear combination of the vectors $\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{r}$ in $R^{n}$ if it can be expressed in the form $$ \mathbf{w}=k_{1} \mathbf{v}_{1}+k_{2} \mathbf{v}_{2}+\cdots+k_{r} \mathbf{v}_{r} $$ where $k_{1}, k_{2}, \ldots, k_{r}$ are scalars. These scalars are called the coefficients of the linear combination. In the case where $r=1$, Formula (14) becomes $\mathbf{w}=k_{1} \mathbf{v}_{1}$, so that a linear combination of a single vector is just a scalar multiple of that vector.Mathematicshttps://mathsgee.com/36707/what-is-a-linear-combination-of-vectors?show=36708#a36708Wed, 26 Jan 2022 00:42:22 +0000Answered: What are the different notations of writing vectors?
https://mathsgee.com/36705/what-are-the-different-notations-of-writing-vectors?show=36706#a36706
<p>Vectors can be written as:</p>
<ol>
<li>comma-delimited: $\mathbf{v}=\left(v_{1}, v_{2}, \ldots, v_{n}\right)$</li>
<li>row vector: \begin{equation}
<br>
\mathbf{v}=\left[\begin{array}{llll}
<br>
v_{1} & v_{2} & \cdots & v_{n}
<br>
\end{array}\right]
<br>
\end{equation}</li>
<li>column vector: \begin{equation}
<br>
\mathbf{v}=\left[\begin{array}{c}
<br>
v_{1} \\
<br>
v_{2} \\
<br>
\vdots \\
<br>
v_{n}
<br>
\end{array}\right]
<br>
\end{equation}</li>
</ol>Mathematicshttps://mathsgee.com/36705/what-are-the-different-notations-of-writing-vectors?show=36706#a36706Wed, 26 Jan 2022 00:40:54 +0000Answered: Give a real life and practical application of linear combinations of vectors
https://mathsgee.com/36703/give-real-practical-application-linear-combinations-vectors?show=36704#a36704
<p>Colors on computer monitors are commonly based on what is called the RGB color model. Colors in this system are created by adding together percentages of the primary colors red $(\mathrm{R})$, green $(\mathrm{G})$, and blue (B). One way to do this is to identify the primary colors with the vectors
<br>
$$
<br>
\begin{array}{ll}
<br>
\mathbf{r}=(1,0,0) & \text { (pure red) } \\
<br>
\mathbf{g}=(0,1,0) & \text { (pure green) } \\
<br>
\mathbf{b}=(0,0,1) & \text { (pure blue) }
<br>
\end{array}
<br>
$$
<br>
in $R^{3}$ and to create all other colors by forming linear combinations of $\mathbf{r}, \mathbf{g}$, and $\mathbf{b}$ using coefficients between 0 and 1 , inclusive; these coefficients represent the percentage of each pure color in the mix.</p>
<p>The set of all such color vectors is called RGB space or the RGB color cube (Figure 3.1.14). Thus, each color vector c in this cube is expressible as a linear combination of the form
<br>
$$
<br>
\begin{aligned}
<br>
\mathbf{c} &=k_{1} \mathbf{r}+k_{2} \mathbf{g}+k_{3} \mathbf{b} \\
<br>
&=k_{1}(1,0,0)+k_{2}(0,1,0)+k_{3}(0,0,1) \\
<br>
&=\left(k_{1}, k_{2}, k_{3}\right)
<br>
\end{aligned}
<br>
$$
<br>
where $0 \leq k_{i} \leq 1 .$ As indicated in the figure, the corners of the cube represent the pure primary colors together with the colors black, white, magenta, cyan, and yellow. The vectors along the diagonal running from black to white correspond to shades of gray.</p>
<p><img alt="cube" src="https://mathsgee.com/?qa=blob&qa_blobid=5483531186152627255" style="height:205px; width:255px"></p>Mathematicshttps://mathsgee.com/36703/give-real-practical-application-linear-combinations-vectors?show=36704#a36704Wed, 26 Jan 2022 00:39:36 +0000