MAT 267 PRACTICE PROBLEMS FOR TEST 1

1. Find the distance from $(2, -5, 3)$ to each of the following. a) The xy-plane b) The yz-plane
   c) The xz-plane d) The x-axis e) The y-axis f) The z-axis
   

2. Write an equation of the set of all points whose distance from the x-axis equals the distance from the point $(1, 2, 3)$ .
   

3. Find equation of the sphere with center $(2, -6, 4)$ and radius 5. Describe its intersection with each of the coordinate planes.
   

4. Find a vector that has the same direction as $<3, -1, 5>$ but has length 7.
   

5. A vector $\textbf{v}$ =$<x, y>$ lies in the second quadrant and makes an angle $\frac{5 \pi}{6}$ with the positive x-axis and its length is $6$ . Find $\textbf{v}$ in component form.
   

6. Three men are trying to hold a ferocious lion still for the veterinarian. The lion, in the center, is wearing a collar with three ropes attached to it and each man has hold of a rope. Charlie is pulling in the direction N60$^\circ$ W with a force of 350 pounds and Sam is pulling in the direction N45$^\circ$ E with a force of 400 pounds. What is the direction and magnitude of the force needed on the third rope to counterbalance Sam and Charlie? (Draw a diagram first).
   

7. For $\mathbf{a}=<1,4,-2>$ and $\mathbf{b}=<2,2,3>$ , find the angle between $\mathbf{a}$ and $\mathbf{b}$ .
   

8. Find a unit vector that is orthogonal to $<4,1>$ . Be sure to justify your answer.
   

9. Suppose you start walking from the origin in the direction of $<-4,3>$ . You make exactly 1 right-angle turn and end up at the point (-2,7). What are the coordinates of the point where you made the turn?
   

10. Find the area of the triangle with vertices $(-1,1,2)$ , $(4,-2,-1)$ , and $(3,0,5)$ .
    

11. Find two unit vectors that are orthogonal to the vectors $\mathbf{a}=\langle 2, -1, 5\rangle$ and $\mathbf{b}=\langle 1, -2, 3\rangle$ .
    

12. Find the angle of intersection of the planes $2x + 3y -z=6$ and $x+2y+2z=7$ . Find the answer in radians and degrees.
    

13. Where does the line $\mathbf{r}(t)=\langle 2, 1,-5\rangle + t\langle 1,-1,2 \rangle$ intersect the plane $2x + 3y -z=6$ ?
    

14. Find an equation of the plane through the points $(1,1,2)$ , $(3,2,2)$ , and $(0,0,4)$ .
    

15. What surface is defined by the equation $-x^2 +4y^2 -z^2 = 4$ ? What plane curves appear on the $y=k$ plane traces? Which axis intersects the surface(s)?
    

16. Reduce the equation $x^2 - 2y^2 = 3z^2$ . to one of the standard forms and identify the surface.
    

17. Name the surface for each equation below.
    • $x^2 - y^2 + z^2 = 1$      $\underline{~~~~~~~~~~}$
    • $x^2+4y^2 +9 z^2 = 1$      $\underline{~~~~~~~~~~}$
    • $y=x^2-z^2$      $\underline{~~~~~~~~~~}$
    • $y=2x^2+z^2$      $\underline{~~~~~~~~~~}$
    • $9x^2+4y^2+z^2=1$      $\underline{~~~~~~~~~~}$
    • $x^2 + 2z^2 = 1$      $\underline{~~~~~~~~~~}$

18. Find the tangent, unit normal and binormal vectors of the function $\mathbf{r}$ (t)= $<\sqrt{2}t, e^{-t}, e^{t}>$ at $t=1$ .
    

19. Find the arc length of the above function between t=1 and 2.
    

20. Find the arc length of the function $\mathbf{r}$ (t)= $<e, e^{2t}, e^{3t}>$ between t=0 and 1. How does this differ from exercise 10.8.3?
    

21. A particle moves along the curve $\mathbf{r}$ (t)= $<1 - t^2, t^3>$ . Calculate the velocity and acceleration vectors for the curve at $t=2$ . Find the parametric equation for the tangent line to the curve at $t=2$ .
    

22. A force $\mathbf{F}$ (t)=$<5,2,-1>$ moves an object along a straight line from the point P $(0,-2,4)$ to Q $(1,2,6)$ . Find the work done by $\mathbf{F}$ .

23. A particle that passes the point P (1,-1) at time t = 0 is moving with velocity $\mathbf{v}$ (t) = $<\cos t , \sin t >$ . Find its speed at $t=4$ and find the parametric equation $\mathbf{r}$ (t) for its motion.
    

SOLUTIONS

1. (a) $3$ (b) $2$ (c) $5$ (d) $\sqrt{34}$ (e) $\sqrt{13}$ (f) $\sqrt{29}$
   

2. $y^2+z^2=(x-1)^2+(y-2)^2+(z-3)^2$ .
   

3. $(x-2)^2+(y+6)^2+(z-4)^2=25$ .

   Intersection with the $xy$ plane is the circle $(x-2)^2+(y+6)^2 = 9$

   It has no intersection with the $xz$ plane

   Intersection with the $yz$ plane is the circle $(y+6)^2+(z-4)^2=21$
   

4. $\frac{7}{\sqrt{35}}<3, -1, 5>$
   

5. $<3\sqrt{2}, 3>$ .
   

6. magnitude = 438.6 pounds, direction S9.28$^\circ$ W
   

7. If $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$ , then
   $$\cos(\theta)=\frac{\mathbf{a}\cdot\mathbf{b}}{\vert\mathbf{a}\vert\vert\mathbf{b}\vert}$$
   $$=\frac{4}{\sqrt{21}\sqrt{17}}$$
   So $\theta=\cos^{-1}\left(\frac{4}{\sqrt{21}\sqrt{17}}\right)\approx77.8^\circ$ .
   

8. Set $<x,y>\cdot<4,1>=0$ . Then any solution to $4x+y=0$ is perpendicular to $<4,1>$ . Pick $x=1$ and $y=-4$ . Then $<1,-4>$ is an orthogonal vector, and a unit vector is $<\frac{1}{\sqrt{17}},\frac{-4}{\sqrt{17}}>$ .
   

9. We need to find the orthogonal projection of $<-2,7>$ onto $<-4,3>$ . This vector is
   $$<-2,7>\cdot<-4,3>\frac{<-4,3>}{\vert<-4,3>\vert^2}=\frac{29}{25}<-4,3>.$$
   Then point of the turn is $<\frac{29(-4)}{25},\frac{29(3)}{25}>$ .
   

10. $\frac{\sqrt{922}}{2}$
    

11. $\frac{1}{\sqrt{59}}<-7, 1, 3>$ and $\frac{1}{\sqrt{59}}<7, -1, -3>$
    

12. 1 rad or 57.68$^\circ$
    

13. $(\frac{2}{3},\frac{7}{3},\frac{-7}{3})$
    

14. $2x-4y-z+4=0$
    

15. Hyperboloid of two sheets. Circles if $k>1$ or $k<-1$ . Intersected by the y-axis.
    

16. $\frac{y^2}{(\frac{1}{\sqrt{2}})^2} + \frac{z^2}{(\frac{1}{\sqrt{3}})^2} = \frac{x^2}{1^2}$ It is a cone.
    

17. • Hyperboloid of one sheet
    • Ellipsoid
    • Hyperbolic paraboloid
    • Elliptic paraboloid
    • Ellipsoid
    • cylinder

18. Solution:
    $$r'(t) = <\sqrt{2}, -e^{-t}, e^{t}>$$
    so
    $$T(t) = \frac{r'(t)}{\vert r'(t)\vert} = \frac{<\sqrt{2}, -e^{-t}... ...}>} {\sqrt{2+e^{-2t}+e^{2t}}} = \frac{<\sqrt{2}, -e^{-t}, e^{t}>} {e^t+e^{-t}}.$$
    The tangent vector at $t=1$ is
    $$T(1) = \frac{<\sqrt{2}, -1/e, e>}{1/e+e} \approx <0.4582, -0.1192, 0.8808>;$$
    
    

    $$T'(t) = <-\frac{\sqrt{2}(e^t-e^{-t})}{(e^t+e^{-t})^2}, \frac{(e^t-e^{-t}... ...^{-t})}, -\frac{(e^t-e^{-t})e^{t}}{(e^t+e^{-t})^2}+\frac{e^{t}}{(e^t+e^{-t})}>$$

    $$= <-\frac{\sqrt{2}(e^t-e^{-t})}{(e^t+e^{-t})^2}, \frac{2}{(e^t+e^{-t})^2}, \frac{2}{(e^t+e^{-t})^2}>$$

    
    Evaluating at $t=1$ ,
    $$T'(1)=\frac{<-\sqrt{2}(e-1/e),2,2>}{(e+1/e)^2}\approx <-0.349, 0.21, 0.21>,$$
    $$N=\frac{T'(1)}{\vert T'(1)\vert}=\frac{<-0.349, 0.21, 0.21>}{\sqrt{-0.349^2+0.21^2+0.21^2}} =<-0.7616, 0.4582, 0.4582>.$$
    $$B=T\times N=<-0.4582, -0.8808, 0.1192>.$$
    
    

19. $$s=\int_1^2 \vert r'(t)\vert dt = \int_1^2 (e^t+e^{-t}) dt = (e^t-e^{-t})\vert _1^2 = 9.6041.$$

20. $$s=\int_0^1 \sqrt{0+4e^{4t}+9e^{6t}} dt= \int_0^1 \left(\frac{(4+9... ...c{3}{2}}{27}\right)'dt = \frac{(4+9e^{2t})^\frac{3}{2}}{27}\vert _0^1 =20.1887.$$
    This is a reparameterization of t in 10.8.3 by $e^t$ and the integration limit here is equivalent to integrating between $t=[1, e]$ in 10.8.3.
    

21. (a) $\mathbf{v}$ (2)= $<-4, 12>$ , $\mathbf{a}$ (2)= $<-2, 12>$ . The parametric equation of the tangent line is $x(t)=-3-4t$ and $y(t)=8+12t$ .
    

22. 12
    

23. speed = 1 and $\mathbf{r}$ (t)= $<\sin t+1, - \cos t>$