Chapter 9

Find the length of the curve $$ x=5+5 t, \quad y=3+4 t, z=t-4 $$ for \(2 \leq t \leq 3\). length \(=\) _________

If \(\mathbf{r}(t)=\cos (3 t) \mathbf{i}+\sin (3 t) \mathbf{j}-8 t \mathbf{k},\) compute: A. The velocity vector \(\mathbf{v}(t)=\) _________ i+ _________\(\mathbf{j}+\)_________\(\mathbf{k}\) \(\mathbf{j}+\)_________\(\mathbf{k}\)

Find the domain of the vector function $$ \mathbf{r}(t)=\left\langle\ln (11 t), \sqrt{t+10}, \frac{1}{\sqrt{12-t}}\right\rangle $$ using interval notation. Domain:________

Rewrite the vector equation \(\mathbf{r}(t)=(-2 t) \mathbf{i}+(3-3 t) \mathbf{j}+(1+3 t) \mathbf{k}\) as the corresponding parametric equations for the line. \(x(t)=\)_________ \(y(t)=\)_________ \(z(t)=\)_________

If \(\mathbf{a}=\mathbf{i}+\mathbf{j}+3 \mathbf{k}\) and \(\mathbf{b}=\mathbf{i}+\mathbf{j}+2 \mathbf{k}\) Compute the cross product \(\mathbf{a} \times \mathbf{b}\). \(\mathbf{a} \times \mathbf{b}=\)_________ \(\mathbf{i}+\)_________\(\mathbf{j}+\) _________\(\mathrm{k}\)

Find \(\mathbf{a} \cdot \mathbf{b}\) if $$ \mathbf{a}=\langle-2,2,-3\rangle \text { and } \mathbf{b}=\langle 4,0,3\rangle $$ \(\mathbf{a} \cdot \mathbf{b}=\)________ Is the angle between the vectors "acute", "obtuse" or "right"?

For each of the following, perform the indicated computation. (a) \((10 \tilde{i}+7 \tilde{j}-5 \tilde{k})-(-6 \tilde{i}+4 \tilde{j}+7 \tilde{k})=\)________ (b) \((10 \tilde{i}+6 \tilde{j}-3 \tilde{k})-2(-3 \tilde{i}+10 \tilde{j}+8 \hat{k})=\)_________

Evaluate a function. Evaluate the function at the specified points. $$ f(x, y)=x+y x^{4},(-1,3),(-4,2),(-1,-3) $$ At (-1,3):_______ At (-4,2):_______ At (-1,-3):_______

Consider the curve \(\mathbf{r}=\left(e^{-5 t} \cos (2 t), e^{-5 t} \sin (2 t), e^{-5 t}\right)\) Compute the arclength function \(s(t):\) (with initial point \(t=0)\).

Given that the acceleration vector is \(\mathbf{a}(t)=(-9 \cos (3 t)) \mathbf{i}+(-9 \sin (3 t)) \mathbf{j}+\) \((3 t) \mathbf{k},\) the initial velocity is \(\mathbf{v}(0)=\mathbf{i}+\mathbf{k},\) and the initial position vector is \(\mathbf{r}(0)=\mathbf{i}+\mathbf{j}+\mathbf{k},\) compute: A. The velocity vector \(\mathbf{v}(t)=\)_________i \(+$$\mathbf{j}+\)_________\(\mathbf{k}\) B. The position vector \(\mathbf{r}(t)=\)_________i \(+ _________$$\mathbf{j}+\)_________\(\mathbf{k}\)

Find a parametrization of the circle of radius 6 in the xy-plane, centered at the origin, oriented clockwise. The point (6,0) should correspond to \(t=0 .\) Use \(t\) as the parameter for all of your answers. \(x(t)=\)_________ \(y(t)=\)_________

Find the vector and parametric equations for the line through the point \(\mathrm{P}(2,-1,-5)\) and parallel to the vector \(-5 \mathbf{i}-3 \mathbf{j}-3 \mathbf{k}\). Vector Form: \(\mathbf{r}=\langle\) _________,_________,-5\rangle\(+t\langle\),_________\(,-3\rangle\) Parametric form (parameter t, and passing through \(\mathrm{P}\) when \(\mathrm{t}=0)\) \(x=x(t)=\)_________ \(y=y(t)=\)_________ \(z=z(t)=\)__________

Suppose \(\vec{v} \cdot \vec{w}=6\) and \(\|\vec{v} \times \vec{w}\|=4,\) and the angle between \(\vec{v}\) and \(\vec{w}\) is \(\theta\) Find (a) \(\tan \theta=\) _______ (b) \(\theta=\)_______-

Determine if the pairs of vectors below are "parallel", "orthogonal", or "neither". $$ \begin{array}{l} \mathbf{a}=\langle-1,-2,2\rangle \text { and } \mathbf{b}=\langle 4,8,10\rangle \text { are } \\ \mathbf{a}=\langle-1,-2,2\rangle \text { and } \mathbf{b}=\langle 4,8,-8\rangle \text { are } \\ \mathbf{a}=\langle-1,-2,2\rangle \text { and } \mathbf{b}=\langle 2,4,-5\rangle \text { are } \end{array} $$

Find a vector a that has the same direction as \langle-6,7,6\rangle but has length \(5 .\) Answer: a = ________

Sketch a contour diagram of each function. Then, decide whether its contours are predominantly lines, parabolas, ellipses, or hyperbolas. (a) \(z=y-2 x^{2}\) (b) \(z=x^{2}+3 y^{2}\) (c) \(z=x^{2}-4 y^{2}\) (d) \(z=-4 x^{2}\)

Find the length of the given curve: $$ \mathbf{r}(t)=(-3 t,-3 \sin t,-3 \cos t) $$ where \(-4 \leq t \leq 1\)

Evaluate $$ \int_{0}^{9}\left(t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\right) d t= $$ _________\(\mathbf{i}+\) _________\(\mathbf{j}+\) _________\(\mathbf{k}\)

Find a vector parametrization of the circle of radius 7 in the xy-plane, centered at the origin, oriented clockwise so that the point (7,0) corresponds to \(t=0\) and the point (0,-7) corresponds to \(t=1\). \(\vec{r}(t)=\)_________

Consider the line which passes through the point \(\mathrm{P}(3,-5,-1),\) and which is parallel to the line \(x=1+6 t, y=2+2 t, z=3+6 t\) Find the point of intersection of this new line with each of the coordinate planes: xy-plane:_________,_________,_________ xz-plane:_________,_________,_________ yz-plane:_________,_________,_________

You are looking down at a map. A vector \(\mathbf{u}\) with \(|\mathbf{u}|=7\) points north and a vector \(\mathbf{v}\) with \(|\mathbf{v}|=6\) points northeast. The crossproduct \(\mathbf{u} \times \mathbf{v}\) points: A) south B) northwest C) up D) down Please enter the letter of the correct answer: _______ The magnitude \(|\mathbf{u} \times \mathbf{v}|=\) _________

Perform the following operations on the vectors \(\vec{u}=\langle 0,5,-4\rangle, \vec{v}=\) \(\langle-2,0,3\rangle,\) and \(\vec{w}=\langle-3,0,1\rangle\). \(\vec{u} \cdot \vec{w}=\) _________ \((\vec{u} \cdot \vec{v}) \vec{u}=\)_________ \(((\vec{w} \cdot \vec{w}) \vec{u}) \cdot \vec{u}=\)_________ \(\vec{u} \cdot \vec{v}+\vec{v} \cdot \vec{w}=\)_________

Let \(\mathbf{a}=<-3,-4,-4>\) and \(\mathbf{b}=<2,2,4>\). Show that there are scalars \(\mathrm{s}\) and \(\mathrm{t}\) so that \(s \mathbf{a}+t \mathbf{b}=<20,24,32>\). You might want to sketch the vectors to get some intuition. \(s=\)_______ \(t=\)_______

Match the surfaces with the verbal description of the level curves by placing the letter of the verbal description to the left of the number of the surface. (a) \(z=\sqrt{\left(x^{2}+y^{2}\right)}\) (b) \(z=2 x+3 y\) (c) \(z=2 x^{2}+3 y^{2}\) (d) \(z=x^{2}+y^{2}\) (e) \(z=x y\) (f) \(z=\frac{1}{x-1}\) \((\mathrm{g}) \quad z=\sqrt{\left(25-x^{2}-y^{2}\right)}\) A. a collection of equally spaced concentric circles B. a collection of unequally spaced concentric circles C. two straight lines and a collection of hyperbolas D. a collection of unequally spaced parallel lines E. a collection of concentric ellipses F. a collection of equally spaced parallel lines

Find the curvature of \(y=\sin (-2 x)\) at \(x=\frac{\pi}{4}\).

Find parametric equations for line that is tangent to the curve \(x=\) \(\cos t, y=\sin t, z=t\) at the point \(\left(\cos \left(\frac{5 \pi}{6}\right), \sin \left(\frac{5 \pi}{6}\right), \frac{5 \pi}{6}\right)\) Parametrize the line so that it passes through the given point at \(\mathrm{t}=0 .\) All three answers are required for credit. \(x(t)=\)_________ \(y(t)=\)_________ \(z(t)=\)_________

Find the point at which the line \(\langle 4,2,4\rangle+t\langle-3,-3,-4\rangle\) intersects the plane \(-5 x+5 y-3 z=2\). _________,_________

If \(\mathbf{a}=\mathbf{i}+8 \mathbf{j}+\mathbf{k}\) and \(\mathbf{b}=\mathbf{i}+10 \mathbf{j}+\mathbf{k},\) find a unit vector with positive first coordinate orthogonal to both a and \(\mathrm{b}\). ________\(\mathbf{i}+\)_________\(\mathbf{j}+$$\mathbf{k}\)

Find \(\mathbf{a} \cdot \mathbf{b}\) if \(|\mathbf{a}|=7,|\mathbf{b}|=7,\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(-\frac{\pi}{10}\) radians. \(\mathbf{a} \cdot \mathbf{b}=\) _________

Resolve the following vectors into components: (a) The vector \(\vec{v}\) in 2 -space of length 5 pointing up at an angle of \(\pi / 4\) measured from the positive \(x\) -axis. \vec{v}=________\(\vec{i}+\)________\(\vec{j}\) (b) The vector \(\vec{w}\) in 3 -space of length 3 lying in the \(x z\) -plane pointing upward at an angle of \(2 \pi / 3\) measured from the positive \(x\) -axis. \vec{v}=________\(\vec{i}+\)________\(\vec{j}\)+________\(\vec{k}\)

The domain of the function \(f(x, y)=\sqrt{x}+\sqrt{y}\) is ________.

Consider the path \(\mathbf{r}(t)=\left(10 t, 5 t^{2}, 5 \ln t\right)\) defined for \(t>0\) Find the length of the curve between the points (10,5,0) and \((40,80,5 \ln (4))\).

If \(\mathbf{r}(t)=\cos (-5 t) \mathbf{i}+\sin (-5 t) \mathbf{j}+6 t \mathbf{k}\) compute \(\mathbf{r}^{\prime}(t)=\) _________\(\mathbf{i}+\) _________\(\mathbf{j}+$$\mathbf{k}\) and \(\int \mathbf{r}(t) d t=\) _________\(\mathbf{i}+\) _________\(\mathbf{j}+\) $$ \mathbf{k}+\mathbf{C} $$ with \(\mathbf{C}\) a constant vector.

Suppose parametric equations for the line segment between (9,-6) and (-2,5) have the form: $$ \begin{array}{l} x=a+b t \\ y=c+d t \end{array} $$ If the parametric curve starts at (9,-6) when \(t=0\) and ends at (-2,5) at \(t=1,\) then find \(a, b, c,\) and \(d\). a=_______,b=________ ,c=________,d=

Find an equation of a plane containing the three points \((5,-2,-2),(2,-5,\) 1), (2,-4,3) in which the coefficient of \(x\) is -9 . ________=0

What is the angle in radians between the vectors $$ \begin{array}{l} \mathbf{a}=(6,-4,9) \text { and } \\ \mathbf{b}=(5,-1,6) ? \end{array} $$ Angle:_______(radians)

Find all vectors \(\vec{v}\) in 2 dimensions having \(\|\vec{v}\|=13\) where the \(\tilde{i}\) -component of \(\vec{v}\) is \(5 \tilde{i}\) vectors: _________ (If you find more than one vector, enter them in a comma-separated list.)

Find the equation of the sphere centered at (10,10,4) with radius \(9 .\) Normalize your equations so that the coefficient of \(x^{2}\) is \(1 .\) _________=0 Give an equation which describes the intersection of this sphere with the plane \(z=5\). _________=0

Find the curvature \(\kappa(t)\) of the curve \(\mathbf{r}(t)=(-3 \sin t) \mathbf{i}+(-3 \sin t) \mathbf{j}+\) \((1 \cos t) \mathbf{k}\)

For the given position vectors \(\mathbf{r}(t)\), compute the (tangent) velocity vector \(\mathbf{r}^{\prime}(t)\) for the given value of \(t\). A) Let \(\mathbf{r}(t)=(\cos t, \sin t)\) Then \(\mathbf{r}^{\prime}\left(\frac{\pi}{4}\right)=\)_________,_________ B) Let \(\mathbf{r}(t)=\) \(\left(t^{2}, t^{3}\right)\) Then \(\mathbf{r}^{\prime}(2)=\) _________,_________ C) Let \(\mathbf{r}(t)=$$e^{t} \mathbf{i}+e^{-2 t} \mathbf{j}+t \mathbf{k}\) Then \(\mathbf{r}^{\prime}(1)=\) _________ $$ \text { i+ } $$_________$$ \mathrm{j}+ $$ _________$$ \text { k ? } $$

Find a parametrization of the curve \(x=-5 z^{2}\) in the xz-plane. Use \(t\) as the parameter for all of your answers. \(x(t)=\)________ \(y(t)=\)________ \(z(t)=\)________

Find an equation for the plane containing the line in the \(x y\) -plane where \(y=1,\) and the line in the \(x z\) -plane where \(z=2\). equation: _________

Let \(A=(5,0,0), B=(2,-2,-3),\) and \(P=(k, k, k) .\) The vector from \(A\) to \(B\) is perpendicular to the vector from \(A\) to \(P\) when \(k=\) ________

Find \(\mathbf{a} \cdot \mathbf{b}\) if \(|\mathbf{a}|=9,|\mathbf{b}|=10,\) and the angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(-\frac{\pi}{3}\) radians. \(\mathbf{a} \cdot \mathbf{b}=\)________

Which is traveling faster, a car whose velocity vector is \(26 \vec{i}+31 \vec{j},\) or a car whose velocity vector is \(40 \vec{i}\), assuming that the units are the same for both directions? \((\square\) the first car \(\square\) the second car) is the faster car. At what speed is the faster car traveling? speed \(=\)_________

(A) If the positive z-axis points upward, an equation for a horizontal plane through the point (-4,1,5) is ___________ (B) An equation for the plane perpendicular to the \(x\) -axis and passing through the point (-4,1,5) is ___________ (C) An equation for the plane parallel to the \(\mathrm{xz}\) -plane and passing through the point (-4,1,5) is ____________

Suppose \(\vec{r}(t)=\cos (\pi t) \boldsymbol{i}+\sin (\pi t) \boldsymbol{j}+3 t \boldsymbol{k}\) represents the position of a particle on a helix, where \(z\) is the height of the particle. (a) What is \(t\) when the particle has height \(6 ?\) t= ______ (b) What is the velocity of the particle when its height is \(6 ?\) \(\vec{v}=\) _______ (c) When the particle has height \(6,\) it leaves the helix and moves along the tangent line at the constant velocity found in part (b). Find a vector parametric equation for the position of the particle (in terms of the original parameter \(t\) ) as it moves along this tangent line. \(L(t)=\) ________

Find parametric equations for the quarter-ellipse from (2,0,9) to (0,-3,9) centered at (0,0,9) in the plane \(z=9\). Use the interval \(0 \leq t \leq \pi / 2\). \(x(t)=\)_________ \(y(t)=\)_________ \(z(t)=\)_________

Find the angle in radians between the planes \(4 x+z=1\) and \(5 y+z=1\).

Find two unit vectors orthogonal to \(\mathbf{a}=\langle-4,-4,5\rangle\) and \(\mathbf{b}=\langle-1,-2,3\rangle\) Enter your answer so that the first non- zero coordinate of the first vector is positive. First Vector: _________,_________ Second Vector:_________._________