Chapter 7

Find symmetric equations for the line through \((4,-11,-7)\) that is parallel to the line \(x=2+5 t, y=-1+\frac{1}{3} t, z=9-2 t\).

Find the angle \(\theta\) between the given vectors. $$ \mathbf{a}=\left\langle\frac{1}{2}, \frac{1}{2}, \frac{3}{2}\right\rangle, \mathbf{b}=\langle 2,-4,6\rangle $$

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

Find the distance from the point \((-6,2,-3)\) to (a) the \(x z\) -plane and (b) the origin.

Find \(\mathbf{a}+(\mathbf{b}+\mathbf{c})\) for the given vectors. \(\mathbf{a}=\langle 1,1\rangle, \mathbf{b}=\langle 4,3\rangle, \mathbf{c}=\langle 0,-2\rangle\)

In Problems \(25-28\), determine whether the given vectors are linearly independent or linearly dependent. $$ \langle 4,-8\rangle,(-6,12) \text { in } R^{2} $$

In Problems \(25-28\), find the direction cosines and direction angles of the given vector. $$ \mathbf{a}=\mathbf{i}+2 \mathbf{j}+3 \mathbf{k} $$

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

The given three points form a triangle. Determine which triangles are isosceles and which are right triangles. $$ (0,0,0),(3,6,-6),(2,1,2) $$

Find a unit vector (a) in the same direction as \(\mathbf{a}\), and (b) in the opposite direction of \(\mathbf{a}\). \(\mathbf{a}=\langle 2,2\rangle\)

In Problems \(25-28\), determine whether the given vectors are linearly independent or linearly dependent. $$ \langle 1,1\rangle,\langle 0,1\rangle,\langle 2,5\rangle \text { in } R^{2} $$

Find parametric equations for the line through \((1,2,8)\) that is (a) parallel to the \(y\) -axis, and (b) perpendicular to the \(x y\) -plane.

Find the direction cosines and direction angles of the given vector. $$ \mathbf{a}=6 \mathbf{i}+6 \mathbf{j}-3 \mathbf{k} $$

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

The given three points form a triangle. Determine which triangles are isosceles and which are right triangles. $$ (0,0,0),(1,2,4),(3,2,2 \sqrt{2}) $$

Find a unit vector (a) in the same direction as \(\mathbf{a}\), and (b) in the opposite direction of \(\mathbf{a}\). \(\mathbf{a}=\langle-3,4\rangle\)

In Problems \(25-28\), determine whether the given vectors are linearly independent or linearly dependent. $$ 1,(x+1),(x+1)^{2} \text { in } P_{2} $$

Show that the lines given by \(\mathbf{r}=t(1,1,1\rangle\) and \(\mathbf{r}=\langle 6,6,6\rangle+\) \(t(-3,-3,-3)\) are the same.

Find the direction cosines and direction angles of the given vector. $$ \mathbf{a}=\langle 1,0,-\sqrt{3}\rangle $$

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

The given three points form a triangle. Determine which triangles are isosceles and which are right triangles. $$ (1,2,3),(4,1,3),(4,6,4) $$

Find a unit vector (a) in the same direction as \(\mathbf{a}\), and (b) in the opposite direction of \(\mathbf{a}\). \(\mathbf{a}=\langle 0,-5\rangle\)

Let \(\mathscr{L}_{a}\) and \(\mathscr{L}_{b}\) be lines with direction vectors a and \(\mathbf{b}\), respectively. \(\mathscr{L}_{a}\) and \(\mathscr{L}_{b}\) areorthogonalifa and \(\mathbf{b}\) are orthogonal and parallel if a and \(\mathbf{b}\) are parallel. Determine which of the following lines are orthogonal and which are parallel. (a) \(\mathbf{r}=\langle 1,0,2\rangle+t\langle 9,-12,6\rangle\) (b) \(x=1+9 t, y=12 t, z=2-6 t\) (c) \(x=2 t, y=-3 t, z=4 t\) (d) \(x=5+t, y=4 t, z=3+\frac{5}{2} t\) (e) \(x=1+t, y=\frac{3}{2} t, z=2-\frac{3}{2} t\) (f) \(\frac{x+1}{-3}=\frac{y+6}{4}=\frac{z-3}{-2}\)

Find the direction cosines and direction angles of the given vector. $$ \mathbf{a}=\langle 5,7,2\rangle $$

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

The given three points form a triangle. Determine which triangles are isosceles and which are right triangles. $$ (1,1,-1),(1,1,1),(0,-1,1) $$

Find a unit vector (a) in the same direction as \(\mathbf{a}\), and (b) in the opposite direction of \(\mathbf{a}\). \(\mathbf{a}=\langle 1,-\sqrt{3}\rangle\)

Explain why \(f(x)=\frac{x}{x^{2}+4 x+3}\) is a vector in \(C[0,3]\) but not a vector in \(C[-3,0]\).

In Problems , determine the points of intersection of the given line and the three coordinate planes. $$ x=4-2 t, y=1+2 t, z=9+3 t $$

Use the distance formula to prove that the given points are collinear. $$ P_{1}(1,2,0), P_{2}(-2,-2,-3), P_{3}(7,10,6) $$

\( \mathbf{a}=\langle 2,8\rangle\) and \(\mathbf{b}=\langle 3,4\rangle .\) Find a unit vector in the same direction as the given vector. \(\mathbf{a}+\mathbf{b}\)

In Problems, determine the points of intersection of the given line and the three coordinate planes. $$ \frac{x-1}{2}=\frac{y+2}{3}=\frac{z-4}{2} $$

A vector space \(V\) on which a dot or inner product has been defined is called an inner product space. An inner product for the vector space \(C[a, b]\) is given by $$ (f, g)=\int_{a}^{b} f(x) g(x) d x $$ In \(C[0,2 \pi]\) compute \((x, \sin x)\).

Show that if two nonzero vectors \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal, then their direction cosines satisfy $$ \cos \alpha_{1} \cos \alpha_{2}+\cos \beta_{1} \cos \beta_{2}+\cos \gamma_{1} \cos \gamma_{2}=0 $$

Use the distance formula to prove that the given points are collinear. $$ P_{1}(2,3,2), P_{2}(1,4,4), P_{3}(5,0,-4) $$

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

\( \mathbf{a}=\langle 2,8\rangle\) and \(\mathbf{b}=\langle 3,4\rangle .\) Find a unit vector in the same direction as the given vector. \(2 \mathbf{a}-3 \mathbf{b}\)

In Problems, determine whether the given lines intersect. If so, find the point of intersection. $$ \begin{aligned} &x=4+t, y=5+t, z=-1+2 t \\ &x=6+2 s, y=11+4 s, z=-3+s \end{aligned} $$

Solve for the unknown. $$ P_{1}(x, 2,3), P_{2}(2,1,1) ; d\left(P_{1}, P_{2}\right)=\sqrt{21} $$

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

Find a vector \(b\) that is parallel to the given vector and has the indicated magnitude. \(\mathbf{a}=3 \mathbf{i}+7 \mathbf{j},\|\mathbf{b}\|=2\)

Find a basis for the solution space of $$ \frac{d^{4} y}{d x^{4}}-2 \frac{d^{3} y}{d x^{3}}+10 \frac{d^{2} y}{d x^{2}}=0 $$

In Problems, determine whether the given lines intersect. If so, find the point of intersection. $$ \begin{aligned} &x=1+t, y=2-t, z=3 t \\ &x=2-s, y=1+s, z=6 s \end{aligned} $$

Determine a unit vector whose direction angles, relative to the three coordinate axes, are equal.

Solve for the unknown. $$ P_{1}(x, x, 1), P_{2}(0,3,5) ; d\left(P_{1}, P_{2}\right)=5 $$

Find a vector \(b\) that is parallel to the given vector and has the indicated magnitude. \(\mathbf{a}=\frac{1}{2} \mathbf{i}-\frac{1}{2} \mathbf{j},\|\mathbf{b}\|=3\)

Let \(\left\\{\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{n}\right\\}\) be any set of vectors in a vector space \(V\). Show that \(\operatorname{Span}\left(\mathbf{x}_{1}, \mathbf{x}_{2}, \ldots, \mathbf{x}_{n}\right)\) is a subspace of \(V\).

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

Find the coordinates of the midpoint of the line segment between the given points. $$ \left(1,3, \frac{1}{2}\right),\left(7,-2, \frac{5}{2}\right) $$

Find a vector in the opposite direction of \(\mathbf{a}=\langle 4,10\rangle\) but \(\frac{3}{4}\) as long.