Practice Problems
1. For what values of a are ~a = (a, −2, 1) and ~b = (2a, a, −4) perpendicular?
2. Find the angles ~a = (3, −6, 2) makes with the coordinate axes.
3. Find the projection of ~a = (1, −2, 1) onto ~b = (4, −4, 7).
4. Find an equation for the plane perpendicular to ~n = (2, 3, 6) and passing through (1, 5, 3).
Find the distance from this plane to the origin.
5. Show that a normal vector to the plane passing through three points with position vectors
p~, ~q, ~r is ~q × p~ + p~ × ~r + ~r × ~q
6. Show that the diagonals of a parallelogram bisect eachother.
7. For the triangle
use vector methods to:
Prove the law of sines,
sin α
a
2
=
sin β
b
2
=
sin γ
c
Prove the law of cosines, b = a + c2 − 2ac cos β.
8. Two sides of a triangle are formed by the vectors ~a = (3, 6, −2) and ~b = (4, −1, 3).
Determine the angles of the triangle.
9. Find the angle between two diagonals of a cube.
10. The force F~ = (1, 2, 3) is acting on a particle which is constrained to the line: `(t) =
(3, 0, 1) + t(−2, 2, 2). What is the force vector in the direction of motion?
11. Find the point on the line passing through (1, −1, 2) and (2, 2, 3) that is closest to the
point (1,1,5). What is the distance?
12. Find the point on the plane x + 2y + 3z = 4 that is closest to (2, 4, 4). What is the
distance?
13. Find an equation of the plane perpendicular to ~n and at a distance r from the origin.
14. Find the area of a parallelogram having diagonals d~1 = (3, 1, −2) and d~2 = (1, −3, 4)
1
15. Practice with the tetrahedron, octehedron, cube.
Position the cube (side length 1) in R3 with a face in the xy-plane and an edge along the
x-axis.
Position the octahedron (side lengths 1) so that two edges are î, ĵ.
Position the tetrahedron (side lengths 1) with a face in the xy-plane and an edge along
the x-axis.
Do things like: Find the vertices, find area of a face, find equation of plane containing a
face, find angle between two faces, parametrize a line containing a given edge, find the
center, lengths of diagonals, the surface area,...
16. Sketch regions (if a variable is not mentioned then it is unconstrained):
(a) 0 ≤ θ ≤ π, 1 ≤ r ≤ 9, 0 ≤ z ≤ 2 (cylindrical coords)
(b) 0 ≤ θ ≤ π, 0 ≤ z ≤ 3 (cylindrical coords)
(c) r2 ≥ z + 1 (cylindrical coords)
(d) ρ > 2, φ = π (spherical coords)
1 2 3
17. Find a function f so that df =
4 5 6
18. Compute
∂F ∂F ∂F
, , ∂z
∂x ∂y
for
(a) F (x, y, z) = (x2 + y 3 , xyz, xyez )
(b) F (x, y, z) = (log(xyz), exy
2 z3
)
19. Compute the differential matrix of:
(a) f (x, y, z) = (x cos y, sin(xy)).
2
2
(b) f (u, v) = (u3 euv , u log v, uv )
(c) f (x, y, z, w) = xyzw.
(d) f (x) = (x, x2 , x3 , x4 )
20. Compute the limits if they exist:
(a) lim(x,y)→(0,0)
x2 −xy+y 2
x2 +y 2
3
(b) lim(x,y)→(0,0) x2x+y2 (this is also can provide a good example of a function whose directional derivatives all exist at (0,0) but is not differentiable at (0,0))
(c) lim(x,y)→(0,0)
x2 +y 2
xy
(d) lim(x,y)→(1,1)
x3 −y 3
x−y
21. What are conditions on the partial derivatives of f : Rn → R at p so that f is differentiable
at p?
2
22. The curve c(t) = (t cos t, t sin t, t2 ) lies on the paraboloid x2 + y 2 = z. For each t find a
vector tangent to the parabaloid and perpendicular to c0 (t) (this vector can depend on t).
23. Show using properties of the gradient that the tangent plane to a graph z = f (x, y) at p
has normal ( ∂f
| , ∂f | , −1).
∂x p ∂y p
24. Compute tangent planes to the given surfaces at the given points:
(a) x2 /4 + y 2 − z 3 = 4 at (2, 2, 1)
(b) log(xyz) = 0 at (1, 1, 1).
(c) x5 − sin(xeyz ) + 5zy 2 cos(y − 2) = 10 at (0,2,1)
(d) y 6 z 3 + cos(πxy 3 ) = log(xyz) at (1,1,1).
25. Find a point on the sphere x2 + y 2 + z 2 = 9 where the tangent plane is parallel to the
tangent plane of z 5 = xy 3 z at (1,1,1).
26. Find a point on z 2 = x3 − y 3 where the tangent plane is parallel to the plane 4x − y − 2z =
10.
Hint: It is easiest to find a point when x is positive and y is negative.
27. We are climbing a hill whose height is given by f (x, y) = 111 − x2 − y 2 and are located
at x = 5, y = 2.
(a) What direction should we set out in to increase our height the fastest?
(b) Compute the directional derivative of f in this direction.
(c) If we climb at a speed of 12 units/s compute how fast our height is changing as we
climb (in the direction of (a)).
(d) We have somewhat followed the optimal path to the summit and ended up at x =
2, y = 1 when we realize we forgot it was tarantula mating season and notice a stampede
of tarantulas tearing down the hill. Now we want to reverse our path in the direction of
quickest descent, which direction should we take?
28. Compute the differential of f ◦ g given:
(a) f (x, y, z) = x2 y 2 ez and g(u, v) = (u4 v − v, v 3 euv , uv) at u = 1, v = 0.
√
(b) f (x, y) = log(x2 + y 2 ), g(u) = (u, 2 − u2 )
29. (a) Find the normal vector ~n to the surface S given by x2 y − 3x + xz + y 2 + 3z = 3 at
x0 = (1, 1, 1).
(b) Find the tangent plane to S at x0 .
(c) Let f (x, y, z) = x22 y + z 3 − 8, find the directional derivative of f in the direction of ~n.
30. Write 120 as a sum of three numbers x, y, z such that the sum of the products (xy+yz+xz)
is a maximum.
31. Find a point on the plane 2x − y + z = 20 closest to (0, 0, 0).
3
32. Show a rectangular box of a given volume V > 0 has minimum surface area when the box
is a cube.
33. Find the point(s) on the surface xyz = 8 which are nearest the origin.
34. Compute the curvature and torsion of the eliptical helix: γ(t) = (a cos t, b sin t, ct) for
some non-zero constants a, b, c.
In the case when a = b = 1 what happens to the curvature as c → ∞? Does this make
geometric sense?
35. What is the torsion of the curve c(t) = (log t5 − sin t2 , cos t1/3 − esin t , 0)?
36. Let f (x, y) = ex cos y. Compute the third order Taylor expansion of f around (0, π).
37. Let f (x, y, z) = ex cos y+z . Compute the second order Taylor expansion of f around (0, 0, 0).
38. Approximate (1.1)3 e0.02 with a second order Taylor approximation.
39. Let f : R → R be a smooth function with f 0 (0) 6= 0.
(a) Compute the 2nd order Taylor expansion of F (x, y) = f (xy) at x = 0, y = 0.
(b) Is (0, 0) a critical point of F ? If so, is it a max/min/saddle?
40. Let f (x, y) = 13 x3 + 12 y 2 + 2xy + 5x + y. Classify all the critical points.
41. Let f (x, y) = x4 + y 4 − x2 − y 2 . Classify all the critical points.
42. Let f (x, y) = x2 + y 4 . Find and classify the critical point.
43. (a) Compute the gradient of log r (where r2 = x2 + y 2 + z 2 ).
(b) Compute the curl and divergence of F~ (x, y, z) = ( x , y , z )
r
r
r
(c) Coompute the curl and divergence of F~ (x, y, z) = (bz − cy, cx − az, ay − bx) (a, b, c
are constants).
4