Chapter 7

Discuss: Is \(R^{2}\) a subspace of \(R^{3} ?\) Are \(R^{2}\) and \(R^{3}\) subspaces of \(R^{4} ?\)

In Problems, determine whether the given lines intersect. If so, find the point of intersection. $$ \begin{aligned} &x=3-t, y=2+t, z=8+2 t \\ &x=2+2 s, y=-2+3 s, z=-2+8 s \end{aligned} $$

In Problems \(33-36, \mathbf{a}=\langle 1,-1,3\rangle\) and \(\mathbf{b}=\langle 2,6,3\rangle .\) Find the indicated number. \(\operatorname{comp}_{\mathrm{a}} \mathbf{b}\)

Find the coordinates of the midpoint of the line segment between the given points. $$ (0,5,-8),(4,1,-6) $$

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

Given that \(\mathbf{a}=\langle 1,1\rangle\) and \(\mathbf{b}=\langle-1,0\rangle\), find a vector in the same direction as \(\mathbf{a}+\mathbf{b}\) but five times as long.

In Problem 9 , you should have proved that the set \(M_{22}\) of \(2 \times 2\) arrays of real numbers $$ M_{22}=\left\\{\left(\begin{array}{ll} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array}\right)\right\\} $$ or matrices, is a vector space with vector addition and scalar multiplication defined in that problem. Find a basis for \(M_{22}\). What is the dimension of \(M_{22} ?\)

The angle between two lines \(\mathscr{L}_{a}\) and \(\mathscr{L}_{b}\) is the angle between their direction vectors a and \(\mathbf{b}\). In Problems, find the angle between the given lines. $$ \begin{aligned} &x=4-t, y=3+2 t, z=-2 t \\ &x=5+2 s, y=1+3 s, z=5-6 s \end{aligned} $$

In Problems \(33-36, \mathbf{a}=\langle 1,-1,3\rangle\) and \(\mathbf{b}=\langle 2,6,3\rangle .\) Find the indicated number. \(\operatorname{comp}_{a}(\mathbf{b}-\mathbf{a})\)

The coordinates of the midpoint of the line segment between \(P_{1}\left(x_{1}, y_{1}, z_{1}\right)\) and \(P_{2}(2,3,6)\) are \((-1,-4,8)\). Find the coordinates of \(P_{1}\).

Find the indicated scalar or vector without using \((5),(13)\), or \((15)\). $$ 2 \mathbf{j} \cdot[\mathbf{i} \times(\mathbf{j}-3 \mathbf{k})] $$

Consider a finite orthogonal set of nonzero vectors \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\\}\) in \(R^{n} .\) Discuss: Is this set linearly independent or linearly dependent?

The angle between two lines \(\mathscr{L}_{a}\) and \(\mathscr{L}_{b}\) is the angle between their direction vectors a and \(\mathbf{b}\). In Problems, find the angle between the given lines. $$ \frac{x-1}{2}=\frac{y+5}{7}=\frac{z-1}{-1} ; \quad \frac{x+3}{-2}=y-9=\frac{z}{4} $$

In Problems \(33-36, \mathbf{a}=\langle 1,-1,3\rangle\) and \(\mathbf{b}=\langle 2,6,3\rangle .\) Find the indicated number. \(\operatorname{comp}_{2 b}(\mathbf{a}+\mathbf{b})\)

Let \(P_{3}\) be the midpoint of the line segment between \(P_{1}(-3,4,1)\) and \(P_{2}(-5,8,3)\). Find the coordinates of the midpoint of the line segment (a) between \(P_{1}\) and \(P_{3}\) and (b) between \(P_{3}\) and \(P_{2}\).

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

In Problems 37 and 38, find the component of the given vector in the direction from the origin to the indicated point. \(\mathbf{a}=4 \mathbf{i}+6 \mathbf{j}, \quad P(3,10)\)

If \(\mathbf{u}, \mathbf{v}\), and \(\mathbf{w}\) are vectors in a vector space \(V\), then the axioms of an inner product \((\mathbf{u}, \mathbf{v})\) are (i) \((\mathbf{u}, \mathbf{v})=(\mathbf{v}, \mathbf{u})\) (ii) \((k \mathbf{u}, \mathbf{v})=k(\mathbf{u}, \mathbf{v}), k\) a scalar (iii) \((\mathbf{u}, \mathbf{u})=0\) if \(\mathbf{u}=\mathbf{0}\) and \((\mathbf{u}, \mathbf{u})>0\) if \(\mathbf{u} \neq \mathbf{0}\) (iv) \((\mathbf{u}, \mathbf{v}+\mathbf{w})=(\mathbf{u}, \mathbf{v})+(\mathbf{u}, \mathbf{w})\) Show that \((\mathbf{u}, \mathbf{v})=u_{1} v_{1}+4 u_{2} v_{2}\), where \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle\) and \(\mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle\), is an inner product on \(R^{2} .\)

Find the vector \(\overrightarrow{P_{1} P_{2}}\). $$ P_{1}(3,4,5), P_{2}(0,-2,6) $$

In Problems 37 and 38, find the component of the given vector in the direction from the origin to the indicated point. \(\mathbf{a}=\langle 2,1,-1\rangle, \quad P(1,-1,1)\)

Find the vector \(\overrightarrow{P_{1} P_{2}}\). $$ P_{1}(-2,4,0), P_{2}\left(6, \frac{3}{2}, 8\right) $$

In Problems \(39-42\), find proj \(_{\mathrm{b}}\). \(\mathbf{a}=-5 \mathbf{i}+5 \mathbf{j}, \mathbf{b}=-3 \mathbf{i}+4 \mathbf{j}\)

In Problems, find an equation of the plane that contains the given point and is perpendicular to the indicated vector. $$ (5,1,3) ; 2 \mathrm{i}-3 \mathrm{j}+4 \mathrm{k} $$

Find the vector \(\overrightarrow{P_{1} P_{2}}\). $$ P_{1}(0,-1,0), P_{2}(2,0,1) $$

Find the indicated scalar or vector. $$ (-\mathbf{a}) \times \mathbf{b} $$

In Problems, find an equation of the plane that contains the given point and is perpendicular to the indicated vector. $$ (1,2,5) ; 4 \mathbf{i}-2 \mathbf{j} $$

Find the vector \(\overrightarrow{P_{1} P_{2}}\). $$ P_{1}\left(\frac{1}{2}, \frac{3}{4}, 5\right), P_{2}\left(-\frac{5}{2},-\frac{9}{4}, 12\right) $$

$$ \mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}=\mathbf{0} $$

In Problems \(39-42\), find proj \(_{\mathrm{b}}\). \(\mathbf{a}=-\mathbf{i}-2 \mathbf{j}+7 \mathbf{k}, \mathbf{b}=6 \mathbf{i}-3 \mathbf{j}-2 \mathbf{k}\)

In Problems, find an equation of the plane that contains the given point and is perpendicular to the indicated vector. $$ (6,10,-7) ;-5 i+3 k $$

\(\mathbf{a}=\langle 1,-3,2\rangle, \mathbf{b}=\langle-1,1,1\rangle\), and \(\mathbf{c}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. \(\mathbf{a}+(\mathbf{b}+\mathbf{c})\)

In Problems, find an equation of the plane that contains the given point and is perpendicular to the indicated vector. $$ (0,0,0) ; 6 \mathbf{i}-\mathbf{j}+3 \mathbf{k} $$

\(\mathbf{a}=\langle 1,-3,2\rangle, \mathbf{b}=\langle-1,1,1\rangle\), and \(\mathbf{c}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. \(2 \mathbf{a}-(\mathbf{b}-\mathbf{c})\)

Express the vector \(\mathbf{a}=2 \mathbf{i}+3 \mathbf{j}\) as a linear combination of the given vectors \(\mathbf{b}\) and \(\mathbf{c}\). \(\mathbf{b}=-2 \mathbf{i}+4 \mathbf{j}, \mathbf{c}=5 \mathbf{i}+7 \mathbf{j}\)

In Problems, find an equation of the plane that contains the given point and is perpendicular to the indicated vector. $$ \left(\frac{1}{2}, \frac{3}{4},-\frac{1}{2}\right) ; 6 \mathrm{i}+8 \mathbf{j}-4 \mathbf{k} $$

\(\mathbf{a}=\langle 1,-3,2\rangle, \mathbf{b}=\langle-1,1,1\rangle\), and \(\mathbf{c}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. \(\mathbf{b}+2(\mathbf{a}-3 \mathbf{c})\)

A vector is said to be tangent to a curve at a point if it is parallel to the tangent line at the point. Find a unit tangent vector to the given curve at the indicated point. \(y=\frac{1}{4} x^{2}+1,(2,2)\)

In Problems 43 and 44, \(\mathbf{a}=4 \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{b}=-\mathbf{i}+\mathbf{j}\). Find the indicated vector. proj \(_{(\mathbf{a}-b)} \mathbf{b}\)

In Problems, find an equation of the plane that contains the given point and is perpendicular to the indicated vector. $$ (-1,1,0) ;-\mathbf{i}+\mathbf{j}-\mathbf{k} $$

\(\mathbf{a}=\langle 1,-3,2\rangle, \mathbf{b}=\langle-1,1,1\rangle\), and \(\mathbf{c}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. \(4(\mathbf{a}+2 \mathbf{c})-6 \mathbf{b}\)

A vector is said to be tangent to a curve at a point if it is parallel to the tangent line at the point. Find a unit tangent vector to the given curve at the indicated point. \(y=-x^{2}+3 x,(0,0)\)

A sled is pulled horizontally over ice by a rope attached to its front. A20-lb force acting at an angle of \(60^{\circ}\) with the horizontal moves the sled \(100 \mathrm{ft}\). Find the work done.

In Problems, find, if possible, an equation of a plane that contains the given points. $$ (3,5,2),(2,3,1),(-1,-1,4) $$

\(\mathbf{a}=\langle 1,-3,2\rangle, \mathbf{b}=\langle-1,1,1\rangle\), and \(\mathbf{c}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. \(\|\mathbf{a}+\mathbf{c}\|\)

Find the work done if the point at which the constant force \(\mathbf{F}=4 \mathbf{i}+3 \mathbf{j}+5 \mathbf{k}\) is applied to an object moves from \(P_{1}(3,1,-2)\) to \(P_{2}(2,4,6)\). Assume \(\|\mathbf{F}\|\) is measured in newtons and \(\|\mathbf{d}\|\) is measured in meters.

In Problems, find, if possible, an equation of a plane that contains the given points. $$ (0,1,0),(0,1,1),(1,3,-1) $$

\(\mathbf{a}=\langle 1,-3,2\rangle, \mathbf{b}=\langle-1,1,1\rangle\), and \(\mathbf{c}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. \(\|\mathbf{c}\|\|2 \mathbf{b}\|\)

In Problems, find, if possible, an equation of a plane that contains the given points. $$ (0,0,0),(1,1,1),(3,2,-1) $$

\(\mathbf{a}=\langle 1,-3,2\rangle, \mathbf{b}=\langle-1,1,1\rangle\), and \(\mathbf{c}=\langle 2,6,9\rangle .\) Find the indicated vector or scalar. \(\left\|\frac{\mathbf{a}}{\|\mathbf{a}\|}\right\|+5\left\|\frac{\mathbf{b}}{\|\mathbf{b}\|}\right\|\)

In Problems, find, if possible, an equation of a plane that contains the given points. $$ (0,0,3),(0,-1,0),(0,0,6) $$