Chapter 7

In Problems, find a vector equation for the line through the given points. $$ (1,2,1),(3,5,-2) $$

In Problems 1-10, determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless stated to the contrary, assume that vector addition and scalar multiplication are the ordinary operations defined on that set. $$ \text { The set of vectors }\left\langle a_{1}, a_{2}\right\rangle, \text { where } a_{1} \geq 0, a_{2} \geq 0 $$

Verify that the basis \(B\) for the given vector space is orthonormal. Use Theorem \(7.7 .1\) to find the coordinates of the vector \(\mathbf{u}\) relative to the basis \(B .\) Then write \(\mathbf{u}\) as a linear combination of the basis vectors. $$ B=\left\\{\left\langle\frac{12}{13}, \frac{5}{13}\right\rangle,\left\langle\frac{5}{13},-\frac{12}{13}\right\rangle\right\\}, \quad R^{2} ; \quad \mathbf{u}=\langle 4,2\rangle $$

In Problems \(1-12, \mathbf{a}=\langle 2,-3,4\rangle, \mathbf{b}=\langle-1,2,5\rangle\), and \(\mathbf{c}=\langle 3,6,-1\rangle .\) Find the indicated scalar or vector. \(\mathbf{a} \cdot \mathbf{b}\)

find \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=\mathbf{i}-\mathbf{j}, \mathbf{b}=\mathbf{3} \mathbf{j}+5 \mathbf{k} $$

Graph the given point. Use the same coordinate axes. $$ (1,1,5) $$

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b},(\mathbf{c}) \mathbf{a}-\mathbf{b},(\mathbf{d})\|\mathbf{a}+\mathbf{b}\|\), and \((e)\|\mathbf{a}-\mathbf{b}\|\) . \(\mathbf{a}=2 \mathbf{i}+4 \mathbf{j}, \mathbf{b}=-\mathbf{i}+4 \mathbf{j}\)

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b}\), (c) \(\mathbf{a}-\mathbf{b}\), (d) \(\|\mathbf{a}+\mathbf{b}\|\), and (e) \(\|\mathbf{a}-\mathbf{b}\|\). $$ \mathbf{a}=2 \mathbf{i}+4 \mathbf{j}, \mathbf{b}=-\mathbf{i}+4 \mathbf{j} $$

In Problems, find a vector equation for the line through the given points. $$ (0,4,5),(-2,6,3) $$

In Problems 1-10, determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless stated to the contrary, assume that vector addition and scalar multiplication are the ordinary operations defined on that set. $$ \text { The set of vectors }\left\langle a_{1}, a_{2}\right\rangle \text { , where } a_{2}=3 a_{1}+1 $$

Verify that the basis \(B\) for the given vector space is orthonormal. Use Theorem \(7.7 .1\) to find the coordinates of the vector \(\mathbf{u}\) relative to the basis \(B .\) Then write \(\mathbf{u}\) as a linear combination of the basis vectors. $$ \begin{aligned} &B=\left\\{\left\langle\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}}\right\rangle,\left\langle 0,-\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}}\right\rangle\right. \\ &\left.\left\langle-\frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}},-\frac{1}{\sqrt{6}}\right\rangle\right\\}, \quad R^{3} ; \quad \mathbf{u}=\langle 5,-1,6\rangle \end{aligned} $$

In Problems \(1-12, \mathbf{a}=\langle 2,-3,4\rangle, \mathbf{b}=\langle-1,2,5\rangle\), and \(\mathbf{c}=\langle 3,6,-1\rangle .\) Find the indicated scalar or vector. \(\mathbf{b} \cdot \mathbf{c}\)

find \(\mathbf{a} \times \mathbf{b}\). $$ a=2 \mathrm{i}+\mathrm{j}, \mathbf{b}=4 \mathbf{i}-\mathbf{k} $$

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b},(\mathbf{c}) \mathbf{a}-\mathbf{b},(\mathbf{d})\|\mathbf{a}+\mathbf{b}\|\), and \((e)\|\mathbf{a}-\mathbf{b}\|\) . \(\mathbf{a}=\langle 1,1\rangle, \mathbf{b}=\langle 2,3\rangle\)

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b}\), (c) \(\mathbf{a}-\mathbf{b}\), (d) \(\|\mathbf{a}+\mathbf{b}\|\), and (e) \(\|\mathbf{a}-\mathbf{b}\|\). $$ \mathbf{a}=\langle 1,1\rangle, \mathbf{b}=\langle 2,3\rangle $$

In Problems, find a vector equation for the line through the given points. $$ \left(\frac{1}{2},-\frac{1}{2}, 1\right),\left(-\frac{3}{2}, \frac{5}{2},-\frac{1}{2}\right) $$

In Problems 1-10, determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless stated to the contrary, assume that vector addition and scalar multiplication are the ordinary operations defined on that set. The set of vectors \(\left\langle a_{1}, a_{2}\right\rangle\), scalar multiplication defined by \(k\left\langle a_{1}, a_{2}\right\rangle=\left\langle k a_{1}, 0\right\rangle\)

Verify that the basis \(B\) for the given vector space is orthogonal. Use Theorem \(7.7 .1\) as an aid in finding the coordinates of the vector \(\mathbf{u}\) relative to the basis \(B\). Then write \(\mathbf{u}\) as a linear combination of the basis vectors. $$ \begin{aligned} &B=\left\\{\langle 1,0,1\rangle,\langle 0,1,0\rangle,\langle-1,0,1\rangle, R^{3}\right. \\ &\mathbf{u}=\langle 10,7,-13\rangle \end{aligned} $$

In Problems \(1-12, \mathbf{a}=\langle 2,-3,4\rangle, \mathbf{b}=\langle-1,2,5\rangle\), and \(\mathbf{c}=\langle 3,6,-1\rangle .\) Find the indicated scalar or vector. \(\mathbf{a} \cdot \mathbf{c}\)

find \(\mathbf{a} \times \mathbf{b}\). $$ \mathrm{a}=\langle 1,-3,1\rangle, \mathrm{b}=(2,0,4) $$

Graph the given point. Use the same coordinate axes. $$ (3,4,0) $$

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b},(\mathbf{c}) \mathbf{a}-\mathbf{b},(\mathbf{d})\|\mathbf{a}+\mathbf{b}\|\), and \((e)\|\mathbf{a}-\mathbf{b}\|\) . \(\mathbf{a}=\langle 4,0\rangle, \mathbf{b}=\langle 0,-5\rangle\)

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b}\), (c) \(\mathbf{a}-\mathbf{b}\), (d) \(\|\mathbf{a}+\mathbf{b}\|\), and (e) \(\|\mathbf{a}-\mathbf{b}\|\). $$ \mathbf{a}=\langle 4,0\rangle, \mathbf{b}=\langle 0,-5\rangle $$

In Problems, find a vector equation for the line through the given points. $$ (10,2,-10),(5,-3,5) $$

In Problems 1-10, determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless stated to the contrary, assume that vector addition and scalar multiplication are the ordinary operations defined on that set. The set of vectors \(\left\langle a_{1}, a_{2}\right\rangle\), where \(a_{1}+a_{2}=0\)

In Problems \(1-12, \mathbf{a}=\langle 2,-3,4\rangle, \mathbf{b}=\langle-1,2,5\rangle\), and \(\mathbf{c}=\langle 3,6,-1\rangle .\) Find the indicated scalar or vector. \(\mathbf{a} \cdot \mathbf{( b}+\mathbf{c})\)

find \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=\langle 1,1,1\rangle, \mathbf{b}=\langle-5,2,3\rangle $$

Graph the given point. Use the same coordinate axes. $$ (6,0,0) $$

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b}\), (c) \(\mathbf{a}-\mathbf{b}\), (d) \(\|\mathbf{a}+\mathbf{b}\|\), and (e) \(\|\mathbf{a}-\mathbf{b}\|\). $$ \mathbf{a}=\frac{1}{6} \mathbf{i}-\frac{1}{6} \mathbf{j}, \mathbf{b}=\frac{1}{2} \mathbf{i}+\frac{5}{6} \mathbf{j} $$

In Problems, find a vector equation for the line through the given points. $$ (1,1,-1),(-4,1,-1) $$

In Problems 1-10, determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless stated to the contrary, assume that vector addition and scalar multiplication are the ordinary operations defined on that set. The set of vectors \(\left\langle a_{1}, a_{2}, 0\right\rangle\)

Use the Gram-Schmidt orthogonalization process (3) to transform the given basis \(B=\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\\}\) for \(R^{2}\) into an orthogonal basis \(B^{\prime}=\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\}\). Then form an orthonormal basis \(B^{\prime \prime}=\left\\{\mathbf{w}_{1}, \mathbf{w}_{2}\right\\}\) (a) First construct \(B^{\prime \prime}\) using \(\mathbf{v}_{1}, \mathbf{u}_{1}\). (b) Then construct \(B^{\prime \prime}\) using \(\mathbf{v}_{1}, \underline{u}_{2}\). (c) Sketch \(B\) and each basis \(B^{\prime \prime}\). $$ B=\\{\langle-3,2\rangle,\langle-1,-1\rangle\\} $$

In Problems \(1-12, \mathbf{a}=\langle 2,-3,4\rangle, \mathbf{b}=\langle-1,2,5\rangle\), and \(\mathbf{c}=\langle 3,6,-1\rangle .\) Find the indicated scalar or vector. \(\mathbf{a} \cdot(4 \mathbf{b})\)

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b},(\mathbf{c}) \mathbf{a}-\mathbf{b},(\mathbf{d})\|\mathbf{a}+\mathbf{b}\|\), and \((e)\|\mathbf{a}-\mathbf{b}\|\) . \(\mathbf{a}=-3 \mathbf{i}+2 \mathbf{j}, \mathbf{b}=\mathbf{7} \mathbf{j}\)

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b}\), (c) \(\mathbf{a}-\mathbf{b}\), (d) \(\|\mathbf{a}+\mathbf{b}\|\), and (e) \(\|\mathbf{a}-\mathbf{b}\|\). $$ \mathbf{a}=-3 \mathbf{i}+2 \mathbf{j}, \mathbf{b}=7 \mathbf{j} $$

In Problems, find a vector equation for the line through the given points. $$ (3,2,1),\left(\frac{5}{2}, 1,-2\right) $$

In Problems 1-10, determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless stated to the contrary, assume that vector addition and scalar multiplication are the ordinary operations defined on that set. The set of vectors \(\left\langle a_{1}, a_{2}\right\rangle\), addition and scalar multiplication defined by $$ \begin{aligned} \left\langle a_{1}, a_{2}\right\rangle+\left\langle b_{1}, b_{2}\right\rangle &=\left\langle a_{1}+b_{1}+1, a_{2}+b_{2}+1\right\rangle \\ k\left(a_{1}, a_{2}\right\rangle &=\left\langle k a_{1}+k-1, k a_{2}+k-1\right\rangle \end{aligned} $$

In Problems \(1-12, \mathbf{a}=\langle 2,-3,4\rangle, \mathbf{b}=\langle-1,2,5\rangle\), and \(\mathbf{c}=\langle 3,6,-1\rangle .\) Find the indicated scalar or vector. \(\mathbf{b} \cdot(\mathbf{a}-\mathbf{c})\)

find \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=4 \mathbf{i}+\mathbf{j}-5 \mathbf{k}, \mathbf{b}=\mathbf{x}+3 \mathbf{j}-\mathbf{k} $$

Graph the given point. Use the same coordinate axes. $$ (5,-4,3) $$

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b}\), (c) \(\mathbf{a}-\mathbf{b}\), (d) \(\|\mathbf{a}+\mathbf{b}\|\), and (e) \(\|\mathbf{a}-\mathbf{b}\|\). $$ \mathbf{a}=\langle 1,3\rangle, \mathbf{b}=-5 \mathbf{a} $$

In Problems 1-10, determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless stated to the contrary, assume that vector addition and scalar multiplication are the ordinary operations defined on that set. The set of real numbers, addition defined by \(\mathbf{x}+\mathbf{y}=x-y\)

In Problems, find parametric equations for the line through the given points. $$ (2,3,5),(6,-1,8) $$

Use the Gram-Schmidt orthogonalization process (3) to transform the given basis \(B=\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\\}\) for \(R^{2}\) into an orthogonal basis \(B^{\prime}=\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}\right\\}\). Then form an orthonormal basis \(B^{\prime \prime}=\left\\{\mathbf{w}_{1}, \mathbf{w}_{2}\right\\}\) (a) First construct \(B^{\prime \prime}\) using \(\mathbf{v}_{1}, \mathbf{u}_{1}\). (b) Then construct \(B^{\prime \prime}\) using \(\mathbf{v}_{1}, \underline{u}_{2}\). (c) Sketch \(B\) and each basis \(B^{\prime \prime}\). $$ B=\\{\langle 1,1\rangle,\langle 1,0\rangle\\} $$

In Problems \(1-12, \mathbf{a}=\langle 2,-3,4\rangle, \mathbf{b}=\langle-1,2,5\rangle\), and \(\mathbf{c}=\langle 3,6,-1\rangle .\) Find the indicated scalar or vector. \(\mathbf{a} \cdot \mathbf{a}\)

find \(\mathbf{a} \times \mathbf{b}\). $$ a=\left\langle\frac{1}{1}, 0, \frac{1}{2}\right\rangle, b=(4,6,0) $$

Describe geometrically all points \(P(x, y, z)\) that satisfy the given condition. $$ z=5 $$

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b},(\mathbf{c}) \mathbf{a}-\mathbf{b},(\mathbf{d})\|\mathbf{a}+\mathbf{b}\|\), and \((e)\|\mathbf{a}-\mathbf{b}\|\) . \(\mathbf{a}=-\mathbf{b}, \mathbf{b}=2 \mathbf{i}-9 \mathbf{j}\)

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b}\), (c) \(\mathbf{a}-\mathbf{b}\), (d) \(\|\mathbf{a}+\mathbf{b}\|\), and (e) \(\|\mathbf{a}-\mathbf{b}\|\). $$ \mathbf{a}=-\mathbf{b}, \mathbf{b}=2 \mathbf{i}-9 \mathbf{j} $$

In Problems 1-10, determine whether the given set is a vector space. If not, give at least one axiom that is not satisfied. Unless stated to the contrary, assume that vector addition and scalar multiplication are the ordinary operations defined on that set. The set of complex numbers \(a+b i\), where \(i^{2}=-1\), addition and scalar multiplication defined by $$ \begin{aligned} \left(a_{1}+b_{1} l\right)+\left(a_{2}+b_{2} i\right) &=\left(a_{1}+a_{2}\right)+\left(b_{1}+b_{2}\right) i \\ k(a+b i) &=k a+k b i, k \text { a real number } \end{aligned} $$