# Cal 3

**Paper, Order, or Assignment Requirements**

- 20pt) For given vectors ~u = h−1, 1, √ 6i and ~v = h2, −2, 0i compute the following. (a) The sum −2~u + ~v, (b) The dot product ~u · ~v, (c) The angle between ~u and ~v. 2 2. (8 pt) a) Find the equation of the line which is parallel to the line ~r(t) = h3t, 1 + 4t, 2 − 5ti and which passes through the point P(−1, 2, −2). (7 pt) b) Find the point (if it exists) at which the line ~r(t) = h2t, 3−3t, −3 + 4ti intersects the plane z = 1. 3 3. (15 pt) A particle is moving in space with acceleration described by the function ~a(t) = ht, t2 , 1i. At the moment t = 0 the instantaneous velocity of the particle is ~v0 = h1, 1, 1i, and the coordinates of the particle are ~r0 = h2, 0, 0i. Find the function ~r(t) describing the trajectory of the particle. 4 4. (20 pt) For the points P(1, 0, 1), Q(2, 1, 1) and R(3, 1, −1), find the following. (a) The area of the parallelogram whose the sides are the vectors −→P Q and −→P R. (b) The equation of the plane containing P, Q and R. 5 5. (10 pt) Find the length of a curve given by ~r(t) = D t, 2 3 (t − 1) 3 2 E , 4 ≤ t ≤ 6 6. (10 pt) Find the equation of the line of the intersection of the planes 2x+y+z = 1 and x−y−z = 5.

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