Math 210 Summer 92 — Calculus III — Exam #1 — Berger — page 1 of 2 Show work on this sheet in 60 minutes ((5 × 10 + 50 = 100)pts).

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1. Let a⃗ =< 2, 3, 1 >, b⃗ =< 1, 4, 5 > c⃗ =< 0, 1, 0 > (a) Find the Volume of the solid whose sides are given by the vectors a⃗, b⃗, and c⃗.

(b) Find the cos(^{}(a⃗, b⃗)):

(c) Find any non-zero vector perpendicular to both a⃗ and b⃗:

2. Let a⃗ be any vector, and u⃗ be a unit vector. Show that: a⃗ − (a⃗ · u⃗) u⃗ is Orthogonal to u⃗.

3. Find the Normal vector and the Vector Equation of the Plane passing through the three points (1, 0, −1), (2, 1, 5) and (2, 1, 7).

4. Find the Vector Equation of the Line passing through the two points (1, 1, 1) and (2, 2, 2).

Math 210 Summer 92 — Calculus III — Exam #1 — Berger — page 2 of 2 5. Reduce to standard form, Classify, and Sketch the Surface, 4x^{2} + y^{2} − 2z^{2} − 2y + 1 = 9.

6. Given the position vector r⃗(t) = sin(3t)i⃗ + tj⃗ + cos(3t)k⃗, find the (a) Velocity:

(b) Acceleration:

(c) Speed:

(d) Unit Tangent Vector:

(e) Arclength for the segment traced out when t varies from π to 2π:

(f) Unit Normal Vector as a function of t:

(g) Curvature as a function of t: