Download Math 206A Exam 01 - Vector Calculus Problems and more Exams Mathematics in PDF only on Docsity!
- Let a =
(2, 5 , 7) and b =
(p, 4 , 1); suppose a × b is
(− 23 , 33 , −17). Find p. Show your work.
- Let
` 1 (t) be the line parameterized by
(2, − 5 , 1) + t
Let the parameteric equations for line
` 2 be x = 4s + 5, y = 7s − 36, and z = 2s + 20.
2A. It’s a fact that these two lines intersect. Find t and s for which
` 1 (t) =
` 2 (s). Show your work.
2B. What is the point of intersection?
2C. In the form
(x − x 0 , y − y 0 , z − z 0 ) · n = 0, what is an equation for the plane containing these two lines?
- Let a =
(3, − 2 , 5) and b =
3a. What is the area of the parallelogram which has vectors a and b as two of its sides?
3b. Find the projection vector projb(a) of a onto b.
3c. Find a unit vector perpendicular to both a and b.
3d. To the nearest 1/10 of a degree, find in degrees the angle θ between a and b.
3e. Let P be the point at the tip of the vector b. What are the cylindrical coordinates of P?
- Consider the sphere centered at (3, − 2 , 5) with radius 7.
6a. What is the equation of this sphere?
6b. What is the equation of the plane tangent to this sphere at its “south pole”?
- Consider an object moving along a path with its position parameterized by g(t) =
(g 1 (t), g 2 (t)) where
g 1 (t) = t^4 −
t^3 − 12 t^2 + 40 and g 2 (t) = t^3 − 3 t.
(Assume t is in seconds and the units on the xy plane are in feet).
7a. Use your calculator to draw a decent, labeled sketch of this path for t ∈ [− 2. 6 , 3 .2]. Set your window to [− 30 , 50] × [− 12 , 25] and set the scales on the x and y axes to 10 and 10.
7b. What are all the times that the horizontal components of the velocity are 0?
7c. Find a parameterization for the line tangent to the path at t = 3.
7d. If the object leaves the path at t = 3 and follows the tangent line with the velocity it has at t = 3, where is it at t = 7?
7e. Bonus! this is a continuation of the problem on the previous page: To the nearest 1/10, when is the object moving slowest, and where is it? Explain!
- Find the equation of the elliptic paraboloid given here: In addition to the two points labeled, it’s tangent to the xy plane. Just FYI, the “contour curves” are at heights z = 2, 4 , 6 , 8....
