MAT 267 TEST 1

1. Find the distance from $(1, -1, 1)$ 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. Find equation of the sphere with center $(1, 1, 1)$ and radius 5. Describe its intersection with each of the coordinate planes.
   

3. Find a vector that has the same direction as $<1, -1, 1>$ but has length 2.
   

4. A vector $\textbf{v}$ =$<x, y>$ lies in the second quadrant and makes an angle $\frac{5 \pi}{4}$ with the positive x-axis and its length is $3$ . Find $x$ and $y$ entries for this vector.
   

5. For $\mathbf{a}=<1,1,-1>$ and $\mathbf{b}=<1,1,1>$ , find cosine of the angle between $\mathbf{a}$ and $\mathbf{b}$ .
   

6. Find a unit vector (vector of magnitude equal to $1)$ that is orthogonal to $<1,1>$ . Be sure to justify your answer.
   

7. Suppose you start walking from the origin in the direction of $<-1,1>$ . 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?
   

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

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

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

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

12. Find an equation of the plane through the points $(1, 1, 1)$ , $(-1,1,2)$ , and $(1,1,4)$ .
    

13. Given a curve defined by the parametrization $\mathbf{r}$ (t)= $<\cos(t),\sin(t), 3t>.$ Calculate its curvature and torsion. Find the arc length of the curve between $t=0$ and $t=\pi.$
    

14. Given a curve defined by the parametrization $\mathbf{r}$ (t)= $<3t -t^3,3t^2, 3t+t^3>.$ Calculate its curvature and torsion.

15. A particle moves along the curve $\mathbf{r}$ (t)= $<1 - t^4, t^2>$ . Calculate the velocity and acceleration vectors for the curve at $t=2$ . Find the parametric equation for the tangent line to the curve at $t=2$ .
    

16. A force $\mathbf{F}$ (t)=$<1,2,-1>$ moves an object along a straight line from the point P $(1,-2,4)$ to Q $(1,-4,3)$ . Find the work done by $\mathbf{F}$ .

17. Calculate volume of the parallelepiped in $4$ -dimensional space spanned by
    

    $$a = (1,\;1,\;1,1)$$

    $$b = (1,\;-1,\;2,-2)$$

    $$c = (1,\;1,\;4,4)$$

    
    

18. Find the distance from $(1, -1, 1)$ to the plane defined by
    $$x_1+x_2+x_3 = 4.$$

19. Find the distance from $(1, 1)$ to the line defined by
    $$x_1+x_2 = 4.$$

20. Find the area of the triangle in $5$ -dimensional space defined by three points

    
    

    $$A = (1,\;1,\;1,1,1)$$

    $$B = (1,\;-1,\;2,-2,3)$$

    $$C = (1,\;1,\;4,4,9)$$