Chapter 7

Describe the locus of points \(P(x, y, z)\) that satisfy the given equation(s). $$ x y z=0 $$

Find the vector \(P_{1} P_{2}\). Graph \(P_{1} P_{2}\) and its corresponding position vector. \(P_{1}(3,2), P_{2}(5,7)\)

In Problems \(11-16\), determine whether the given set is a subspace of the vector space \(C(-\infty, \infty)\). 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. $$ (-5,-2,-4),(1,1,2) $$

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

Determine a scalar \(c\) so that the given vectors are orthogonal. (a) \(\mathbf{a}=2 \mathbf{i}-c \mathbf{j}+3 \mathbf{k}, \mathbf{b}=3 \mathbf{i}+2 \mathbf{j}+4 \mathbf{k}\) (b) \(\mathbf{a}=\left\langle c, \frac{1}{2}, c\right\rangle, \mathbf{b}=\langle-3,4, c\rangle\)

Describe the locus of points \(P(x, y, z)\) that satisfy the given equation(s). $$ x^{2}+y^{2}+z^{2}=0 $$

Find the vector \(P_{1} P_{2}\). Graph \(P_{1} P_{2}\) and its corresponding position vector. \(P_{1}(-2,-1), P_{2}(4,-5)\)

In Problems \(17-20\), determine whether the given set is a subspace of the indicated vector space. Polynomials of the form \(p(x)=c_{3} x^{3}+c_{1} x ; P_{3}\)

In Problems, find symmetric equations for the line through the given points. $$ (5,10,-2),(5,1,-14) $$

An inner product defined on the vector space \(P_{2}\) of all polynomials of degree less than or equal to 2 , is given by $$ (p, q)=\int_{-1}^{1} p(x) q(x) d x $$ Use the Gram-Schmidt orthogonalization process to transform the given basis \(B\) for \(P_{2}\) into an orthogonal basis \(B^{\prime}\). $$ B=\left\\{1, x, x^{2}\right\\} $$

Find a vector \(\mathbf{v}=\left\langle x_{1}, y_{1}, 1\right\rangle\) that is orthogonal to both \(\mathbf{a}=\langle 3,1,-1\rangle\) and \(\mathbf{b}=\langle-3,2,2\rangle\)

Describe the locus of points \(P(x, y, z)\) that satisfy the given equation(s). $$ (x+1)^{2}+(y-2)^{2}+(z+3)^{2}=0 $$

Find the vector \(P_{1} P_{2}\). Graph \(P_{1} P_{2}\) and its corresponding position vector. \(P_{1}(3,3), P_{2}(5,5)\)

In Problems \(17-20\), determine whether the given set is a subspace of the indicated vector space. Polynomials \(p\) that are divisible by \(x-2 ; P_{2}\)

In Problems, find symmetric equations for the line through the given points. $$ \left(\frac{5}{6},-\frac{1}{4}, \frac{1}{5}\right),\left(\frac{1}{3}, \frac{3}{8},-\frac{1}{10}\right) $$

An inner product defined on the vector space \(P_{2}\) of all polynomials of degree less than or equal to 2 , is given by $$ (p, q)=\int_{-1}^{1} p(x) q(x) d x $$ Use the Gram-Schmidt orthogonalization process to transform the given basis \(B\) for \(P_{2}\) into an orthogonal basis \(B^{\prime}\). $$ B=\left\\{x^{2}-x, x^{2}+1,1-x^{2}\right\\} $$

A rhombus is an oblique-angled parallelogram with all four sides equal. Use the dot product to show that the diagonals of a rhombus are perpendicular.

Describe the locus of points \(P(x, y, z)\) that satisfy the given equation(s). $$ (x-2)(z-8)=0 $$

Find the vector \(P_{1} P_{2}\). Graph \(P_{1} P_{2}\) and its corresponding position vector. \(P_{1}(0,3), P_{2}(2,0)\)

In Problems, find parametric and symmetric equations for the line through the given point parallel to the given vector. $$ (4,6,-7), \mathbf{a}=\left\langle 3, \frac{1}{2},-\frac{3}{2}\right\rangle $$

Verify that the vector $$ \mathbf{c}=\mathbf{b}-\frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\|^{2}} \mathbf{a} $$ is orthogonal to the vector \(\mathbf{a}\).

Find the indicated scalar or vector without using \((5),(13)\), or \((15)\). $$ (2 \mathbf{i}) \times \mathbf{j} $$

Describe the locus of points \(P(x, y, z)\) that satisfy the given equation(s). $$ z^{2}-25=0 $$

Find the terminal point of the vector \(P_{1} P_{2}=4 \mathbf{i}+8 \mathbf{j}\) if its initial point is \((-3,10)\).

In Problems \(17-20\), determine whether the given set is a subspace of the indicated vector space. Functions \(f\) such that \(\int_{a}^{b} f(x) d x=0 ; C[a, b]\)

In Problems, find parametric and symmetric equations for the line through the given point parallel to the given vector. $$ (1,8,-2), \mathbf{a}=-7 \mathbf{i}-8 \mathbf{j} $$

Determine a scalar \(c\) so that the angle between \(\mathbf{a}=\mathbf{i}+c \mathbf{j}\) and \(\mathbf{b}=\mathbf{i}+\mathbf{j}\) is \(45^{\circ}\).

Find the indicated scalar or vector without using \((5),(13)\), or \((15)\). $$ \mathbf{i} \times(-3 \mathbf{k}) $$

Describe the locus of points \(P(x, y, z)\) that satisfy the given equation(s). $$ x=y=z $$

Find the initial point of the vector \(P_{1} P_{2}=\langle-5,-1\rangle\) if its terminal point is \(\langle 4,7\rangle\).

Find the initial point of the vector \(P_{1} P_{2}^{\prime}=\langle-5,-1\rangle\) if its terminal point is \(\langle 4,7\rangle\).

In 3-space, a line through the origin can be written as \(S=\\{(x, y, z) \mid x=a t, y=b t, z=c t, a, b, c\) real numbers \(\\} .\) With addition and scalar multiplicationthe same as for vectors \(\langle x, y, z\rangle\), show that \(S\) is a subspace of \(R^{3}\).

In Problems, find parametric and symmetric equations for the line through the given point parallel to the given vector. $$ (0,0,0), \mathbf{a}=5 \hat{i}+9 \mathbf{j}+4 \mathbf{k} $$

In Problems 21-24, find the angle \(\theta\) between the given vectors. $$ \mathbf{a}=3 \mathbf{i}-\mathbf{k}, \mathbf{b}=2 \mathbf{i}+2 \mathbf{k} $$

Find the distance between the given points. $$ (3,-1,2),(6,4,8) $$

Determine which of the following vectors are parallel to \(\mathbf{a}=4 \mathbf{i}+6 \mathbf{j}\) (a) \(-4 \mathbf{i}-6 \mathbf{j}\) (b) \(-\mathbf{i}-\frac{3}{2} \mathbf{j}\) (c) \(10 \mathbf{i}+15 \mathbf{j}\) (d) \(2(\mathbf{i}-\mathbf{j})-3\left(\frac{1}{2} \mathbf{i}-\frac{5}{12} \mathbf{j}\right)\) (e) \(8 \mathbf{i}+12 \mathbf{j}\) (f) \((5 \mathbf{i}+\mathbf{j})-(7 \mathbf{i}+4 \mathbf{j})\)

The vectors \(\mathbf{u}_{1}=\langle 1,0,0\rangle, \mathbf{u}_{2}=\langle 1,1,0\rangle\), and \(\mathbf{u}_{3}=\langle 1,1,1\rangle\) form a basis for the vector space \(R^{3}\). (a) Show that \(\mathbf{u}_{1}, \mathbf{u}_{2}\), and \(\mathbf{u}_{3}\) are linearly independent. (b) Express the vector \(\mathbf{a}=\langle 3,-4,8\rangle\) as a linear combination of \(\mathbf{u}_{1}, \mathbf{u}_{2}\), and \(\mathbf{u}_{3}\).

In Problems, find parametric and symmetric equations for the line through the given point parallel to the given vector. $$ (0,-3,10), \mathbf{a}=\langle 12,-5,-6\rangle $$

Find the angle \(\theta\) between the given vectors. $$ \mathbf{a}=2 \mathbf{i}+\mathbf{j}, \mathbf{b}=-3 \mathbf{i}-4 \mathbf{j} $$

Find the indicated scalar or vector without using \((5),(13)\), or \((15)\). $$ \mathbf{i} \times(\mathbf{j} \times \mathbf{k}) $$

Find the distance between the given points. $$ (-1,-3,5),(0,4,3) $$

Determine a scalar \(c\) so that \(\mathbf{a}=3 \mathbf{i}+c \mathbf{j}\) and \(\mathbf{b}=-\mathbf{i}+9 \mathbf{j}\) are parallel.

The vectors \(\mathbf{u}_{1}=\langle 1,0,0\rangle, \mathbf{u}_{2}=\langle 1,1,0\rangle\), and \(\mathbf{u}_{3}=\langle 1,1,1\rangle\) form a basis for the vector space \(R^{3}\). (a) Show that \(\mathbf{u}_{1}, \mathbf{u}_{2}\), and \(\mathbf{u}_{3}\) are linearly independent. (b) Express the vector \(\mathbf{a}=\langle 3,-4,8\rangle\) as a linear combination of \(\mathbf{u}_{1}, \mathbf{u}_{2}\), and \(\mathbf{u}_{3}\).

Find parametric equations for the line through \((6,4,-2)\) that is parallel to the line \(x / 2=(1-y) / 3=(z-5) / 6\).

Find the angle \(\theta\) between the given vectors. $$ \mathbf{a}=\langle 2,4,0\rangle, \mathbf{b}=\langle-1,-1,4\rangle $$

The set of vectors \(\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\\}\), where $$ \mathbf{u}_{1}=\langle 1,1,3\rangle, \mathbf{u}_{2}=\langle 1,4,1\rangle, \text { and } \mathbf{u}_{3}=\langle 1,10,-3\rangle $$ is linearly dependent in \(R^{3}\) since \(\mathbf{u}_{3}=-2 \mathbf{u}_{1}+3 \mathbf{u}_{2} .\) Discuss what you would expect when the Gram-Schmidt process in (4) is applied to these vectors. Then carry out the orthogonalization process.

Find the distance from the point \((7,-3,-4)\) to \((\) a) the \(y z\) -plane and (b) the \(x\) -axis.

Find \(\mathbf{a}+(\mathbf{b}+\mathbf{c})\) for the given vectors. \(\mathbf{a}=\langle 5,1\rangle, \mathbf{b}=\langle-2,4\rangle, \mathbf{c}=\langle 3,10\rangle\)

The vectors \(p_{1}(x)=x+1, p_{2}(x)=x-1\) form a basis for the vector space \(P_{1}\) (a) Show that \(p_{1}(x)\) and \(p_{2}(x)\) are linearly independent. (b) Express the vector \(p(x)=5 x+2\) as a linear combination of \(p_{1}(x)\) and \(p_{2}(x)\).