MATH 210  EXAM 1  FALL 1992

NAME: (PRINT) SIGNATURE: SSN: Show your work in the space provided.

1. Let u⃗ = ⟨1, 3, −2⟩ and v⃗ = ⟨4, 2, 1⟩.

a) Find the cosine of the angle θ between u⃗ and v⃗.

b) Find the length of the projection of v⃗ on u⃗.

2. Reduce to standard form, classify, and sketch the surface x^{2} − 4x − 6y^{2} + z^{2} + 2 = 0.

3. Find a vector equation of the line through (2, 1, 3) and (4, 5, 6). Then write the parametric equations of the line.

MATH 210  EXAM 1  FALL 1992

4. Let P = (2, 0, 3), Q = (1, 2, 0), and R = (0, 0, 1).

a) Find the equation of the plane through the points P, Q, and R.

b) Find the distance from the point S = (0, 3, 0) to the plane in part a).

5. Consider the curve given by

r⃗(t) = cos t i⃗ + sin t j⃗ + 4t k⃗.

a) Find the arc length from t = \frac{π}{2} to t = π.

b) Reparametrize the curve by arc length measured from the point where t = 0.

MATH 210  EXAM 1  FALL 1992

6. Given the position vector

r⃗(t) = sin(2t)i⃗ + cos(3t)j⃗ + 2t k⃗

find the following:

a) Velocity =

b) Acceleration =

c) Speed =

d) Unit tangent vector =

e) Curvature =

f) Unit normal vector =