Math 210 Midterm I February 12, 1993

Name SS number

CIRCLE YOUR ANSWER SHOW ALL WORK

1) (10pt) Find the cosine of the angle between the vectors ⟨1, 2, −3⟩ and ⟨−1, 0, 2⟩.

2) (10pt) Find the projection of 2i⃗ − 3j⃗ + k⃗ onto i⃗ + 6j⃗ − 2k⃗.

3) (10 pt) Find parametric equations for the line through the points (2,-7,5) and (-4,2,5).

4) (10pt) Find the equation of the plane passing through the points (-1,1,1), (1,-1,2) and (4,0,3).

5) (12 pt) Find parametric equations for the line of intersection of the planes x+3y −z = 1 and x − y + z = 2.

6) (12 pt) A particle moves through space with acceleration a⃗(t) = ⟨2, sin t, cos t⟩, initial velocity v⃗(0) = ⟨0, 0, 0⟩ and initial position r⃗(0) = ⟨0, 0, 0⟩. Find the position vector r⃗(t).

7) (12 pt) Let r⃗(t) = ⟨sin πt, \sqrt{t}, cos πt⟩. Find parametric equations for the line tangent to the curve parameterized by r⃗ at the point (0,1,-1).

8) (12 pt) Let r⃗(t) = ⟨t, e^{t}, 1⟩. For t = 0 find the unit tangent vector T(t) and the unit normal vector N(t).

9) (12 pt) Let r⃗(t) = ⟨t^{3}, t^{2}, t⟩. Find the curvature κ(t).