Chapter 7

In Problems, find parametric equations for the line through the given points. $$ (2,0,0),(0,4,9) $$

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 \(B^{\prime \prime}\). $$ B=\\{\langle 5,7\rangle,\langle 1,-2\rangle\\} $$

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. \((2 b) \cdot(3 c)\)

Describe geometrically all points \(P(x, y, z)\) that satisfy the given condition. $$ x=1 $$

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b},(\mathbf{c}) \mathbf{a}-\mathbf{b},(\mathbf{d})\|\mathbf{a}+\mathbf{b}\|\), and \((e)\|\mathbf{a}-\mathbf{b}\|\) . \(\mathbf{a}=\langle 7,10\rangle, \mathbf{b}=\langle 1,2\rangle\)

Find (a) \(3 \mathbf{a}\), (b) \(\mathbf{a}+\mathbf{b}\), (c) \(\mathbf{a}-\mathbf{b}\), (d) \(\|\mathbf{a}+\mathbf{b}\|\), and (e) \(\|\mathbf{a}-\mathbf{b}\|\). $$ \mathbf{a}=(7,10), \quad \mathbf{b}=(1,2) $$

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. The set of arrays of real numbers \(\left(\begin{array}{ll}a_{11} & a_{12} \\\ a_{21} & a_{22}\end{array}\right)\), addition and scalar multiplication defined by \(\begin{aligned}\left(\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right)+\left(\begin{array}{ll}b_{11} & b_{12} \\ b_{21} & b_{22}\end{array}\right)=\left(\begin{array}{ll}a_{12}+b_{12} & a_{11}+b_{11} \\\ a_{22}+b_{22} & a_{21}+b_{21}\end{array}\right) \\ & k\left(\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right)=\left(\begin{array}{ll}k a_{11} & k a_{12} \\ k a_{21} & k a_{22}\end{array}\right) \end{aligned}\)

In Problems, find parametric equations for the line through the given points. $$ (1,0,0),(3,-2,-7) $$

Use the Gram-Schmidt orthogonalization process (4) to transform the given basis \(\boldsymbol{B}=\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\\}\) for \(R^{3}\) into an orthogonal basis \(B^{\prime}=\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\\}\). Then form an orthonormal basis \(B^{\prime \prime}=\left\\{\mathbf{w}_{1}, \mathbf{w}_{2}, \mathbf{w}_{3}\right\\}\) $$ \boldsymbol{B}=\\{\langle 1,1,0\rangle,\langle 1,2,2\rangle,\langle 2,2,1\rangle\\} $$

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. \(\mathbf{a} \cdot(\mathbf{a}+\mathbf{b}+\mathbf{c})\)

find \(\mathbf{a} \times \mathbf{b}\). $$ a=(2,2,-4), b=(-3,-3,6) $$

Describe geometrically all points \(P(x, y, z)\) that satisfy the given condition. $$ x=2, y=3 $$

Find (a) \(4 \mathrm{a}-2 \mathrm{~b}\) and \((\mathrm{b})-3 \mathrm{a}-5 \mathrm{~b}\). \(\mathbf{a}=\langle 1,-3\rangle, \mathbf{b}=\langle-1,1\rangle\)

In Problems, find parametric equations for the line through the given points. $$ (0,0,5),(-2,4,0) $$

Use the Gram-Schmidt orthogonalization process (4) to transform the given basis \(\boldsymbol{B}=\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\\}\) for \(R^{3}\) into an orthogonal basis \(B^{\prime}=\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\\}\). Then form an orthonormal basis \(B^{\prime \prime}=\left\\{\mathbf{w}_{1}, \mathbf{w}_{2}, \mathbf{w}_{3}\right\\}\) $$ B=\\{\langle-3,1,1\rangle,\langle 1,1,0\rangle,\langle-1,4,1\rangle\\} $$

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. \((2 a) \cdot(a-2 b)\)

find \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=(8,1,-6), \mathbf{b}=\langle 1,-2,10\rangle $$

Describe geometrically all points \(P(x, y, z)\) that satisfy the given condition. $$ x=4, y=-1, z=7 $$

Find (a) \(4 \mathrm{a}-2 \mathrm{~b}\) and \((\mathrm{b})-3 \mathrm{a}-5 \mathrm{~b}\). \(\mathbf{a}=\mathbf{i}+\mathbf{j}, \mathbf{b}=3 \mathbf{i}-2 \mathbf{j} \quad\)

In Problems \(11-16\), determine whether the given set is a subspace of the vector space \(C(-\infty, \infty)\). All functions \(f\) such that \(f(1)=0\)

In Problems, find parametric equations for the line through the given points. $$ \left(4, \frac{1}{2}, \frac{1}{3}\right),\left(-6,-\frac{1}{4}, \frac{1}{6}\right) $$

Use the Gram-Schmidt orthogonalization process (4) to transform the given basis \(\boldsymbol{B}=\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\\}\) for \(R^{3}\) into an orthogonal basis \(B^{\prime}=\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\\}\). Then form an orthonormal basis \(B^{\prime \prime}=\left\\{\mathbf{w}_{1}, \mathbf{w}_{2}, \mathbf{w}_{3}\right\\}\) $$ B=\left\\{\left\langle\frac{1}{2}, \frac{1}{2}, 1\right\rangle,\left\langle-1,1,-\frac{1}{2}\right\rangle,\left\langle-1, \frac{1}{2}, 1\right\rangle\right\\} $$

find \(\overrightarrow{P_{1} P_{2}} \times \overrightarrow{P_{1} P_{3}}\) $$ P_{1}(2,1,3), P_{2}(0,3,-1), P_{3}(-1,2,4) $$

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. \(\left(\frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}}\right) \mathbf{b}\)

Give the coordinates of the vertices of the rectangular parallelepiped whose sides are the coordinate planes and the planes \(x=2, y=5, z=8\)

Find (a) \(4 \mathrm{a}-2 \mathrm{~b}\) and \((\mathrm{b})-3 \mathrm{a}-5 \mathrm{~b}\). \(\mathbf{a}=\mathbf{i}-\mathbf{j}, \mathbf{b}=-3 \mathbf{i}+4 \mathbf{j}\)

In Problems \(11-16\), determine whether the given set is a subspace of the vector space \(C(-\infty, \infty)\). All functions \(f\) such that \(f(0)=1\)

In Problems, find parametric equations for the line through the given points. $$ (-3,7,9),(4,-8,-1) $$

Use the Gram-Schmidt orthogonalization process (4) to transform the given basis \(\boldsymbol{B}=\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\\}\) for \(R^{3}\) into an orthogonal basis \(B^{\prime}=\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\\}\). Then form an orthonormal basis \(B^{\prime \prime}=\left\\{\mathbf{w}_{1}, \mathbf{w}_{2}, \mathbf{w}_{3}\right\\}\) $$ B=\\{\langle 1,1,1\rangle,\langle 9,-1,1\rangle,\langle-1,4,-2\rangle\\} $$

find \(\overrightarrow{P_{1} P_{2}} \times \overrightarrow{P_{1} P_{3}}\) $$ P_{1}(0,0,1), P_{2}(0,1,2), P_{3}(1,2,3) $$

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. \((\mathbf{c} \cdot \mathbf{b}) \mathbf{a}\)

Find (a) \(4 \mathrm{a}-2 \mathrm{~b}\) and \((\mathrm{b})-3 \mathrm{a}-5 \mathrm{~b}\). \(\mathbf{a}=\langle 2,0\rangle, \mathbf{b}=\langle 0,-3\rangle\)

In Problems \(11-16\), determine whether the given set is a subspace of the vector space \(C(-\infty, \infty)\). All nonnegative functions \(f\)

In Problems, find symmetric equations for the line through the given points. $$ (1,4,-9),(10,14,-2) $$

The given vectors span a subspace \(W\) of \(R^{3}\). Use the Gram-Schmidt orthogonalization process to construct an orthonormal basis for the subspace. $$ \mathbf{u}_{1}=\langle 1,5,2\rangle, \mathbf{u}_{2}=\langle-2,1,1\rangle $$

In Problems 13 and 14 , find \(\mathbf{a} \cdot \mathbf{b}\) if the smaller angle between a and \(\mathbf{b}\) is as given. $$ \|\mathbf{a}\|=10, \quad\|\mathbf{b}\|=5, \quad \theta=\pi / 4 $$

Find a vector that is perpendicular to both a and b. $$ \mathbf{a}=\mathbf{2} \mathbf{i}+\mathbf{7} \mathbf{j}-\mathbf{4} \mathbf{k}, \mathbf{b}=\mathbf{i}+\mathbf{j}-\mathbf{k} $$

Consider the point \(P(-2,5,4)\). (a) If lines are drawn from \(P\) perpendicular to the coordinate planes, what are the coordinates of the point at the base of each perpendicular? (b) If a line is drawn from \(P\) to the plane \(z=-2\), what are the coordinates of the point at the base of the perpendicular? (c) Find the point in the plane \(x=3\) that is closest to \(P\).

Find (a) \(4 \mathrm{a}-2 \mathrm{~b}\) and \((\mathrm{b})-3 \mathrm{a}-5 \mathrm{~b}\). \(\mathbf{a}=\langle 4,10\rangle, \mathbf{b}=-2\langle 1,3\rangle\)

In Problems \(11-16\), determine whether the given set is a subspace of the vector space \(C(-\infty, \infty)\). All functions \(f\) such that \(f(-x)=f(x)\)

In Problems, find symmetric equations for the line through the given points. $$ \left(\frac{2}{3}, 0,-\frac{1}{4}\right),\left(1,3, \frac{1}{4}\right) $$

The given vectors span a subspace \(W\) of \(R^{3}\). Use the Gram-Schmidt orthogonalization process to construct an orthonormal basis for the subspace. $$ \mathbf{u}_{1}=\langle 1,2,3\rangle, \mathbf{u}_{2}=\langle 3,4,1\rangle $$

Find a vector that is perpendicular to both a and b. $$ \mathbf{a}=(-1,-2,4), \mathbf{b}=\langle 4,-1,0\rangle $$

Determine an equation of a planeparallel to a coordinate plane that contains the given pair of points. (a) \((3,4,-5),(-2,8,-5)\) (b) \((1,-1,1),(1,-1,-1)\) (c) \((-2,1,2),(2,4,2)\)

Find (a) \(4 \mathrm{a}-2 \mathrm{~b}\) and \((\mathrm{b})-3 \mathrm{a}-5 \mathrm{~b}\). \(\mathbf{a}=\langle 3,1\rangle+\langle-1,2\rangle, \mathbf{b}=\langle 6,5\rangle-\langle 1,2\rangle\)

$$ \mathbf{a}=\langle 3,1\rangle+\langle-1,2\rangle, \mathbf{b}=\langle 6,5\rangle-\langle 1,2\rangle $$

In Problems \(11-16\), determine whether the given set is a subspace of the vector space \(C(-\infty, \infty)\). All differentiable functions \(f\) All functions \(f\) of the form \(f(x)=c_{1} e^{x}+c_{2} x e^{x}\)

In Problems, find symmetric equations for the line through the given points. $$ (4,2,1),(-7,2,5) $$

The given vectors span a subspace \(W\) of \(R^{4}\). Use the Gram-Schmidt orthogonalization process to construct an orthonormal basis for the subspace. $$ \mathbf{u}_{1}=\langle 1,-1,1,-1\rangle, \mathbf{u}_{2}=\langle 1,3,0,1\rangle $$

Determine which pairs of the following vectors are orthogonal: (a) \(\langle 2,0,1\rangle\) (b) \(3 \mathbf{i}+2 \mathbf{j}-\mathbf{k}\) (c) \(2 \mathbf{i}-\mathbf{j}-\mathbf{k}\) (d) \(\mathbf{i}-4 \mathbf{j}+6 \mathbf{k}\) (e) \(\langle 1,-1,1\rangle\) (f) \(\langle-4,3,8\rangle\)