# Recent questions tagged product

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What is the dot (Euclidean inner) product of two vectors?
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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?
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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? ...
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Find the dot product of the vectors shown in the diagram below
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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 &nbsp; ...
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What is the component form of the dot product?
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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?
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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?
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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 $R^{4}$ : $$\mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0)$$
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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 $R^{4}$ : $$\mathbf{u}=(-1,3,5,7), \quad \mathbf{v}=(-3,-4,1,0)$$ ...
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Find the angle between a diagonal of a cube and one of its edges
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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?
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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 \ ...
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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 ...
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When are two nonzero vectors orthogonal to each other?
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When are two nonzero vectors orthogonal to each other?When are two nonzero vectors orthogonal to each other? ...
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