Review for Test 1: Chapters 13 and 14

1)  Given the vectors v = < 2, 5, -3 > and w = < 2, 1, -1 >, find the following:

a)   v w

b)  proj _w v, the vector projection of v onto w

c)  |w|

2. Given the same two vectors v = < 2, 5, -3 > and w = < 2, 1, -1 >, find the following:

a)   v x w

b)  two unit vectors orthogonal to both v and w

3)  Find the parametric and symmetric equations for the line through the point P(2, -3, 5) that is parallel to .

4) Consider the points P₁ (-1, 1, 1), P₂ (4, 3, 7) and P₃ (3, 1, 0). Find an equation for the plane that contains all three points. Simplify completely. Find a vector that is normal to the plane in part a)

5) Answer the following questions about cylindrical and spherical coordinates:

a) Change (2, , 3) from cylindrical coordinates to rectangular coordinates.

b) Change (4, ) from spherical coordinates to rectangular coordinates.

c) Find the distance between the points in part (a) and (b).

6)  Describe in words the surface whose equation is . Include name, and if appropriate, center point, radius, axis it is parallel to, etc.

7)  Given that r(t) = find:

a)   r(t)

b)  Find r’(t)

8)  Find the length of the curve of r(t) = t²i + 2tj + ln(t)k for 0 < t < e.

9)  Given that r(t) = , find:

a)   The unit tangent vector T

b)  The unit normal vector N

c)  The curvature 

10. Given that the acceleration of a particle is a(t) = (2t + 3)²i + 4t³j and v(0) = 3i + j, find the speed of the particle at t=1.