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Calculus 3 problems. Help me solve the problems with full work.


1.

Sketch on a set of xyz − axes and describe the following: x 2 + y 2 + z 2 < 16.

Use the two points listed below for the next three questions. P(-2, -3, 4) Q(-6, 4, 3)

2. Find the vector of PQ

3. Find the magnitude of PQ

4. Find the unit vector in the direction of PQ.

5. Find the dot product u⋅v for u=3i−2j+k and v=i−2j+k.

6. Determine if u = 〈−4, 5〉 and v = 〈2, 6〉 are orthogonal. If the vectors are not orthogonal, keep vector u the same, but change vector v to make the vectors orthogonal. (Do not use v = 〈0, 0〉 to make the vectors orthogonal if they are orthogonal.)

7. Find the vector projection projv u for v = i − 2j + 4k and u = 2i − 2j + 4k.

8. Find the angle formed by a=i−2j+4k and b=2i−2j+k. (Round your answer to two decimal places. You may use degrees or radians. However, label your answer.)

9. Find the area of the parallelogram with the vertices of (1, 1, 1), (2, 2, 2), (2, 3, 4), and (3, 4, 5).

10. Find the equation of the plane through the points (1, 2, 3), (2, 4, 6), and (−1, 0, 2).

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