MAT 267 PRACTICE PROBLEMS FOR TEST 1

1. Find the distance from 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 .

3. Find equation of the sphere with center and radius 5. Describe its intersection with each of the coordinate planes.

4. Find a vector that has the same direction as but has length 7.

5. A vector = lies in the second quadrant and makes an angle with the positive x-axis and its length is . Find 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 W with a force of 350 pounds and Sam is pulling in the direction N45 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 and , find the angle between and .

8. Find a unit vector that is orthogonal to . Be sure to justify your answer.

9. Suppose you start walking from the origin in the direction of . 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 , , and .

11. Find two unit vectors that are orthogonal to the vectors and .

12. Find the angle of intersection of the planes and . Find the answer in radians and degrees.

13. Where does the line intersect the plane ?

14. Find an equation of the plane through the points , , and .

15. What surface is defined by the equation ? What plane curves appear on the plane traces? Which axis intersects the surface(s)?

16. Reduce the equation . to one of the standard forms and identify the surface.

17. Name the surface for each equation below.

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18. Find the tangent, unit normal and binormal vectors of the function (t)= at .

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

20. Find the arc length of the function (t)= between t=0 and 1. How does this differ from exercise 10.8.3?

21. A particle moves along the curve (t)= . Calculate the velocity and acceleration vectors for the curve at . Find the parametric equation for the tangent line to the curve at .

22. A force (t)= moves an object along a straight line from the point P to Q . Find the work done by .

23. A particle that passes the point P (1,-1) at time t = 0 is moving with velocity (t) = . Find its speed at and find the parametric equation (t) for its motion.

SOLUTIONS

1. (a) (b) (c) (d) (e) (f)

2. .

3. .

Intersection with the plane is the circle

It has no intersection with the plane

Intersection with the plane is the circle

4. .

5. magnitude = 438.6 pounds, direction S9.28 W

6. If is the angle between and , then

So .

7. Set . Then any solution to is perpendicular to . Pick and . Then is an orthogonal vector, and a unit vector is .

8. We need to find the orthogonal projection of onto . This vector is

Then point of the turn is .

9. and

11. Hyperboloid of two sheets. Circles if or . Intersected by the y-axis.

12. It is a cone.

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

13. Solution:

so

The tangent vector at is

Evaluating at ,

14. This is a reparameterization of t in 10.8.3 by and the integration limit here is equivalent to integrating between in 10.8.3.

15. (a) (2)= , (2)= . The parametric equation of the tangent line is and .

16. 12

17. speed = 1 and (t)=