CARNEGIE MELLON UNIVERSITY 21-259 Calculus in 3-D Department of Mathematics Spring 2003

Review Problems for Test 1

1. Let a^{⇀}=< 1, 3, −1 >, b^{⇀}=< 0, 4, 2 >, c^{⇀}=< 5, −1, 3 > . Compute each of the following:

(a) |a^{⇀}|

(b) a^{⇀} · c^{⇀}

(c) the angle between a^{⇀} and c^{⇀}

(d) the vector projection of b^{⇀} onto a^{⇀}

(e) the direction cosines of c^{⇀}

(f) a^{⇀} ·( b^{⇀} × c^{⇀})

2. (a) Write an equation for the plane which passes through (3, −6, 8) and is

normal to < 7, −2, −3 > .

(b) Write the symmetric and parametric equations of the line which passes through (1, 3, −5) and is parallel to < 7, −2, −3 > .

3. Find the line of intersection of the planes x + y + z = 4 and 2x − y + 5z = −1.

4. Find a number d such that the distance from the point P = (4, 3, 0) to the plane x − 2y + z = d is 3.

5. Find the distance from the point (1, 0, −1) to the line x = t, y = −1 − t, z = 1 + 2t.

6. Let a^{⇀}=< 2, 0, −1 > and b^{⇀}=< 1, 2, 0 > . Write b^{⇀} as the sum of a vector b^{⇀}_{1} parallel to a^{⇀} and a vector b^{⇀}_{2} perpendicular to a^{⇀} .

7. Find the points of intersection of the line \frac{x}{−2} = \frac{y−1}{3} = \frac{z}{\sqrt{3}} and the sphere x^{2} + 3x + y^{2} + z^{2} = 2.

8. Find the plane that contains the line x = y = z and the point P_{0} = (1, 2, 3).

9. Given that a^{⇀} and b^{⇀} are vectors with |a^{⇀}| = 1 and |a^{⇀} − b^{⇀}| = | b^{⇀}| , find a^{⇀} · b^{⇀} .

10. Let a^{⇀}=< 1, 1, 1 >, b^{⇀}=< 2, −3, 1 > and c^{⇀}=< 1, 0, 2 > . Find a vector v^{⇀} satis-

fying

a^{⇀} · v^{⇀}= b^{⇀} · v^{⇀}= 0

c^{⇀} · v^{⇀}= 2.

11. Let v^{⇀} be a vector satisfying |v^{⇀} × i^{⇀}| = A, |v^{⇀} × j^{⇀}| = B, |v^{⇀} × k^{⇀}| = C Express

|v^{⇀}| in terms of A, B, C.

12. (a) A space curve is described by a vector function r^{⇀} (t) satisfying

r′^{⇀} (t) =< e^{t} cos 2t, e^{t} sin 2t, e^{t} > .

Find the unit tangent vector T^{⇀} (t), the principal unit normal vector N^{⇀} (t), the binormal vector B⃗(t), and the curvature κ(t).

(b) The torsion τ of a space curve is defined by τ = −N⃗ · \frac{dB⃗}{dS}.

Find the torsion of the curve from part (a) when t = 0.

13. Find the length of the curve described by

r^{⇀} (t) =< sin 3t, cos 3t, 2t^{3/2} >

for 1 ≤ t ≤ 3.

14. Find the limit, or show that the limit does not exist.

(a) lim_{(x,y)→(0,0)} \frac{x^{2} + y^{2} + 2x^{4}}{x^{2} + y^{2}}
(b) lim_{(x,y)→(0,0)} \frac{x^{3}}{xy^{2} + y^{3}}

15. Suppose that f (x, y) = (y − x^{2})^{2} . Sketch the level curves of f for c = 0 and c = 1.

16. Suppose that f (x, y) = sin (x + 3y) + x^{2}y^{3} + 2y^{2}. Compute f_{x}, f_{y}, f_{xx}, f_{xy}, and f_{yy}.