# Recent questions tagged product

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What is the dot (Euclidean inner) product of two vectors?
What is the dot (Euclidean inner) product of two vectors?What is the dot (Euclidean inner) product of two vectors? ...
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Knowing the dot product of two vectors, how can I establish the angle between them?
Knowing the dot product of two vectors, how can I establish the angle between them?Knowing the dot product of two vectors, how can I establish the angle between them? ...
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Find the dot product of the vectors shown in the diagram below
Find the dot product of the vectors shown in the diagram below Find the dot product of the vectors shown in the diagram below &nbsp; ...
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What is the component form of the dot product?
What is the component form of the dot product?What is the component form of the dot product? ...
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Which mathematician came up with the dot product notation?
Which mathematician came up with the dot product notation?Which mathematician came up with the dot product notation? ...
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How do I calculate the dot product of two vectors?
How do I calculate the dot product of two vectors?How do I calculate the dot product of two vectors? ...
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Calculate $\mathbf{u} \cdot \mathbf{v}$ for the following vectors in $R^{4}$ : $$\mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0)$$
Calculate $\mathbf{u} \cdot \mathbf{v}$ for the following vectors in $R^{4}$ : $$\mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0)$$Calculate $\mathbf{u} \cdot \mathbf{v}$ for the following vectors in $R^{4}$ : $$\mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0)$$ ...
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Find the angle between a diagonal of a cube and one of its edges
Find the angle between a diagonal of a cube and one of its edgesFind the angle between a diagonal of a cube and one of its edges ...
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What are the algebraic properties of the dot/inner product?
What are the algebraic properties of the dot/inner product?What are the algebraic properties of the dot/inner product? ...
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Prove that if $\mathbf{u}$ and $\mathbf{v}$ are vectors in $R^{n}$ with the Euclidean inner product, then $$\mathbf{u} \cdot \mathbf{v}=\frac{1}{4}\|\mathbf{u}+\mathbf{v}\|^{2}-\frac{1}{4}\|\mathbf{u}-\mathbf{v}\|^{2}$$Prove that if $\mathbf{u}$ and $\mathbf{v}$ are vectors in $R^{n}$ with the Euclidean inner product, then $$\mathbf{u} \cdot \mathbf{v}=\frac{1}{4}\| ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If the product of two numbers is 1050 and their $\mathrm{HCF}$ is 25 , find their LCM. 1 answer 10 views If the product of two numbers is 1050 and their $\mathrm{HCF}$ is 25 , find their LCM.If the product of two numbers is 1050 and their $\mathrm{HCF}$ is 25 , find their LCM. ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 List the prime factors of 60 0 answers 7 views List the prime factors of 60List the prime factors of 60 ... close 0 answers 13 views close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let $A$ be a positive definite $n \times n$ real matrix, $\vec{b}$ a real vector, and $\vec{N}$ a real unit vector. 1 answer 7 views Let $A$ be a positive definite $n \times n$ real matrix, $\vec{b}$ a real vector, and $\vec{N}$ a real unit vector.Let $A$ be a positive definite $n \times n$ real matrix, $\vec{b}$ a real vector, and $\vec{N}$ a real unit vector. a) For which value(s) of t ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let $\vec{x}$ and $\vec{p}$ be points in $\mathbb{R}^{n}$. Under what conditions on the scalar $c$ is the set $\|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0$ 0 answers 8 views Let $\vec{x}$ and $\vec{p}$ be points in $\mathbb{R}^{n}$. Under what conditions on the scalar $c$ is the set $\|\vec{x}\|^{2}+2\langle\vec{p}, \vec{x}\rangle+c=0$a) Let $\vec{x}$ and $\vec{p}$ be points in $\mathbb{R}^{n}$. Under what conditions on the scalar $c$ is the set \ \|\vec{x}\|^{2}+2\langle\v ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find the function $f \in \operatorname{span}\{1 \sin x, \cos x\}$ that minimizes $\|\sin 2 x-f(x)\|$, where the norm comes from the inner product 0 answers 9 views Find the function $f \in \operatorname{span}\{1 \sin x, \cos x\}$ that minimizes $\|\sin 2 x-f(x)\|$, where the norm comes from the inner productFind the function $f \in \operatorname{span}\{1 \sin x, \cos x\}$ that minimizes $\|\sin 2 x-f(x)\|$, where the norm comes from the inner product ... close 0 answers 13 views Let $C[-1,1]$ be the real inner product space consisting of all continuous functions $f:[-1,1] \rightarrow \mathbb{R}$, with the inner product $\langle f, g\rangle:=\int_{-1}^{1} f(x) g(x) d x$.Let $C-1,1$ be the real inner product space consisting of all continuous functions $f:-1,1 \rightarrow \mathbb{R}$, with the inner product $\ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(\mathcal{P}_{2}$ be the space of polynomials $p(x)=a+b x+c x^{2}$ of degree at most 2 with the inner product $\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x$. 0 answers 6 views Let $\mathcal{P}_{2}$ be the space of polynomials $p(x)=a+b x+c x^{2}$ of degree at most 2 with the inner product $\langle p, q\rangle=\int_{-1}^{1} p(x) q(x) d x$.Let $\mathcal{P}_{2}$ be the space of polynomials $p(x)=a+b x+c x^{2}$ of degree at most 2 with the inner product $\langle p, q\rangle=\int_{-1}^ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(\mathcal{P}_{2}$ be the space of quadratic polynomials. 0 answers 10 views Let $\mathcal{P}_{2}$ be the space of quadratic polynomials.Let $\mathcal{P}_{2}$ be the space of quadratic polynomials. a) Show that $\langle f, g\rangle=f(-1) g(-1)+f(0) g(0)+f(1) g(1)$ is an inner produ ... close 0 answers 14 views Using the inner product of the previous problem, let $\mathcal{B}=\left\{1, x, 3 x^{2}-1\right\}$ be an orthogonal basis for the space $\mathcal{P}_{2}$ of quadratic polynomials and . . .Using the inner product of the previous problem, let $\mathcal{B}=\left\{1, x, 3 x^{2}-1\right\}$ be an orthogonal basis for the space $\mathcal{P} ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Using the inner product \(\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x$, for which values of the real constants $\alpha, \beta, \gamma$ are the quadratic polynomials ... 1 answer 8 views Using the inner product $\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x$, for which values of the real constants $\alpha, \beta, \gamma$ are the quadratic polynomials ...Using the inner product $\langle f, g\rangle=\int_{-1}^{1} f(x) g(x) d x$, for which values of the real constants $\alpha, \beta, \gamma$ are the ... close 0 answers 10 views In a complex vector space (with a hermitian inner product), if a matrix $A$ satisfies $\langle X, A X\rangle=0$ for all vectors $X$, show that $A=0$. [The previous problem shows that this is false in a real vector space].In a complex vector space (with a hermitian inner product), if a matrix $A$ satisfies $\langle X, A X\rangle=0$ for all vectors $X$, show that \ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Proof or counterexample. Here $v, w, z$ are vectors in a real inner product space $H$. 0 answers 9 views Proof or counterexample. Here $v, w, z$ are vectors in a real inner product space $H$.Proof or counterexample. Here $v, w, z$ are vectors in a real inner product space $H$. a) Let $v, w, z$ be vectors in a real inner product space ... close 0 answers 9 views Let $A$ be a positive definite $n \times n$ real matrix, $b \in \mathbb{R}^{n}$, and consider the quadratic polynomial $Q(x):=\frac{1}{2}\langle x, A x\rangle-\langle b, x\rangle$Let $A$ be a positive definite $n \times n$ real matrix, $b \in \mathbb{R}^{n}$, and consider the quadratic polynomial \ Q(x):=\frac{1}{2}\lang ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let $\ell$ be any linear functional. Show there is a unique vector $v \in \mathbb{R}^{n}$ so that $\ell(x):=\langle x, v\rangle$. 0 answers 6 views Let $\ell$ be any linear functional. Show there is a unique vector $v \in \mathbb{R}^{n}$ so that $\ell(x):=\langle x, v\rangle$.LINEAR FUNCTIONALS In $R^{n}$ with the usual inner product, a linear functional $\ell:$ $\mathbb{R}^{n} \rightarrow \mathbb{R}$ is just a line ... close 0 answers 9 views close 0 answers 7 views Let $V$ be the real vector space of continuous real-valued functions on the closed interval $[0,1]$, and let $w \in V$. For $p, q \in V$, define $\langle p, q\rangle=\int_{0}^{1} p(x) q(x) w(x) d x$.Let $V$ be the real vector space of continuous real-valued functions on the closed interval $0,1$, and let $w \in V$. For $p, q \in V$, defi ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let $U, V, W$ be orthogonal vectors and let $Z=a U+b V+c W$, where $a, b, c$ are scalars. 0 answers 11 views Let $U, V, W$ be orthogonal vectors and let $Z=a U+b V+c W$, where $a, b, c$ are scalars.Let $U, V, W$ be orthogonal vectors and let $Z=a U+b V+c W$, where $a, b, c$ are scalars. a) (Pythagoras) Show that $\|Z\|^{2}=a^{2}\|U\|^{2}+b ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Find all vectors in the plane (through the origin) spanned by \(\mathbf{V}=(1,1-2)$ and $\mathbf{W}=(-1,1,1)$ that are perpendicular to the vector $\mathbf{Z}=(2,1,2)$. 0 answers 11 views Find all vectors in the plane (through the origin) spanned by $\mathbf{V}=(1,1-2)$ and $\mathbf{W}=(-1,1,1)$ that are perpendicular to the vector $\mathbf{Z}=(2,1,2)$.Find all vectors in the plane (through the origin) spanned by $\mathbf{V}=(1,1-2)$ and $\mathbf{W}=(-1,1,1)$ that are perpendicular to the vector ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let $V, W$ be vectors in $\mathbb{R}^{n}$. 0 answers 7 views Let $V, W$ be vectors in $\mathbb{R}^{n}$.Let $V, W$ be vectors in $\mathbb{R}^{n}$. a) Show that the Pythagorean relation $\|V+W\|^{2}=\|V\|^{2}+\|W\|^{2}$ holds if and only if $V$ a ... close 0 answers 8 views Let $A$ be an $m \times n$ matrix, and suppose $\vec{v}$ and $\vec{w}$ are orthogonal eigenvectors of $A^{T} A$. Show that $A \vec{v}$ and $A \vec{w}$ are orthogonal.Let $A$ be an $m \times n$ matrix, and suppose $\vec{v}$ and $\vec{w}$ are orthogonal eigenvectors of $A^{T} A$. Show that $A \vec{v}$ and ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let $W$ be a linear space with an inner product and $A: W \rightarrow W$ be a linear map whose image is one dimensional (so in the case of matrices, it has rank one). 1 answer 15 views Let $W$ be a linear space with an inner product and $A: W \rightarrow W$ be a linear map whose image is one dimensional (so in the case of matrices, it has rank one).Let $W$ be a linear space with an inner product and $A: W \rightarrow W$ be a linear map whose image is one dimensional (so in the case of matrice ... close 0 answers 17 views Let $\vec{v}$ and $\vec{w}$ be vectors in $\mathbb{R}^{n}$. If $\|\vec{v}\|=\|\vec{w}\|$, show there is an orthogonal matrix $R$ with $R \vec{v}=\vec{w}$ and $R \vec{w}=\vec{v}$.Let $\vec{v}$ and $\vec{w}$ be vectors in $\mathbb{R}^{n}$. If $\|\vec{v}\|=\|\vec{w}\|$, show there is an orthogonal matrix $R$ with $R \v ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(\mathcal{P}_{2}$ be the space of polynomials of degree at most 2 . 0 answers 12 views Let $\mathcal{P}_{2}$ be the space of polynomials of degree at most 2 .Let $\mathcal{P}_{2}$ be the space of polynomials of degree at most 2 . &nbsp; a) Find a basis for this space. b) Let $D: \mathcal{P}_{2} \righta ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(\mathcal{P}_{1}$ be the linear space of real polynomials of degree at most one, so a typical element is $p(x):=a+b x$, where $a$ and $b$ are real numbers. 0 answers 8 views Let $\mathcal{P}_{1}$ be the linear space of real polynomials of degree at most one, so a typical element is $p(x):=a+b x$, where $a$ and $b$ are real numbers.Let $\mathcal{P}_{1}$ be the linear space of real polynomials of degree at most one, so a typical element is $p(x):=a+b x$, where $a$ and $b$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 If $A$ and $B$ are $4 \times 4$ matrices such that $\operatorname{rank}(A B)=3$, then $\operatorname{rank}(B A)<4$. 0 answers 8 views If $A$ and $B$ are $4 \times 4$ matrices such that $\operatorname{rank}(A B)=3$, then $\operatorname{rank}(B A)<4$.For each of the following, answer TRUE or FALSE. If the statement is false in even a single instance, then the answer is FALSE. There is no need to ju ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a linear map. Show that the following are equivalent. 0 answers 8 views Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a linear map. Show that the following are equivalent.Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a linear map. Show that the following are equivalent. a) For every $y \in \mathbb{R}^{k}$ th ... close 0 answers 11 views Let $\mathcal{P}_{k}$ be the space of polynomials of degree at most $k$ and define the linear map $L: \mathcal{P}_{k} \rightarrow \mathcal{P}_{k+1}$ by $L p:=p^{\prime \prime}(x)+x p(x) .$Let $\mathcal{P}_{k}$ be the space of polynomials of degree at most $k$ and define the linear map $L: \mathcal{P}_{k} \rightarrow \mathcal{P}_{k+ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What are the rules of differentiation? 1 answer 8 views What are the rules of differentiation?What are the rules of differentiation? ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Verify that the Cauchy-Schwarz inequality holds. \(\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$ 0 answers 23 views Verify that the Cauchy-Schwarz inequality holds. $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$Verify that the Cauchy-Schwarz inequality holds. (b) $\mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)$ ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What is the dot product of \mathbf{u} a column matrix and \mathbf{v} a column matrix? 1 answer 12 views What is the dot product of \mathbf{u} a column matrix and \mathbf{v} a column matrix?What is the dot product of \mathbf{u} a column matrix and \mathbf{v} a column matrix? ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What is the dot product of u a row matrix and v a column matrix? 1 answer 14 views What is the dot product of u a row matrix and v a column matrix?What is the dot product of u a row matrix and v a column matrix? ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What is the dot product of u a column matrix and v a row matrix? 1 answer 13 views What is the dot product of u a column matrix and v a row matrix?What is the dot product of u a column matrix and v a row matrix? ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Verify that the Cauchy–Schwarz inequality holds for \mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1) 0 answers 12 views Verify that the Cauchy–Schwarz inequality holds for \mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1)Verify that the Cauchy&amp;ndash;Schwarz inequality holds for \mathbf{u}=(0,2,2,1), \mathbf{v}=(1,1,1,1) ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Verify that the Cauchy–Schwarz inequality holds for \mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3) 0 answers 18 views Verify that the Cauchy–Schwarz inequality holds for \mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3)Verify that the Cauchy&amp;ndash;Schwarz inequality holds for \mathbf{u}=(4,1,1), \mathbf{v}=(1,2,3) ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Verify that the Cauchy–Schwarz inequality holds for \mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2) 0 answers 16 views Verify that the Cauchy–Schwarz inequality holds for \mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2)Verify that the Cauchy&amp;ndash;Schwarz inequality holds for \mathbf{u}=(1,2,1,2,3), \mathbf{v}=(0,1,1,5,-2) ... close 0 answers 49 views Let \mathbf{r}_{0}=\left(x_{0}, y_{0}\right) be a fixed vector in R^{2}. In each part, describe in words the set of all vectors \mathbf{r}=(x, y) that satisfy the stated condition.Let \mathbf{r}_{0}=\left(x_{0}, y_{0}\right) be a fixed vector in R^{2}. In each part, describe in words the set of all vectors \mathbf{r}=(x, y) ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Show that two nonzero vectors \mathbf{v}_{1} and \mathbf{v}_{2} in R^{3} are orthogonal if and only if their direction cosines satisfy 0 answers 14 views Show that two nonzero vectors \mathbf{v}_{1} and \mathbf{v}_{2} in R^{3} are orthogonal if and only if their direction cosines satisfyShow that two nonzero vectors \mathbf{v}_{1} and \mathbf{v}_{2} in R^{3} are orthogonal if and only if their direction cosines satisfy$$ \cos \ ...
Let $\mathbf{u}$ be a vector in $R^{100}$ whose $i$ th component is $i$, and let $\mathbf{v}$ be the vector in $R^{100}$ whose $i$ th component is $1 /(i+1)$. Find the dot product of $\mathbf{u}$ and $\mathbf{v}$.Let $\mathbf{u}$ be a vector in $R^{100}$ whose $i$ th component is $i$, and let $\mathbf{v}$ be the vector in $R^{100}$ whose $i$ th component is \$1 ...