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Allen – Math 265 B2 Practice Exam #1 Name:
- Given that ~u = 〈− 1 , 3 〉, ~v = 〈 2 , 1 〉, and w~ = 〈 5 , 6 〉 are three non-collinear vectors, find scalars a and b so
that w~ = a~u + b~v.
- Let ~u = 〈 1 , 4 , 2 〉, ~v = 〈− 2 , 3 , 1 〉, and w~ = 〈 4 , − 1 , − 1 〉. Perform the indicated operations.
(a) Compute the angle (in degrees) between ~u and ~v.
(b) Find the volume of the parallelepiped determined by ~u, ~v, and w~.
- Let ~u = 〈 3 , 4 〉 and ~v = 〈 5 , 12 〉. We can write ~u = m~ + ~n, where m~ is parallel to ~v and ~n is perpendicular
to ~v. Find m~ and ~n.
- Find the equation of the plane that passes through the points P = (1, 4 , 6), Q = (− 2 , 5 , −1), and
R = (1, − 1 , 1).
- Find the center and radius of the sphere 4x
2
2
2 − 24 x + 56y − 40 z − 197 = 0.
- A particle P travels in the plane, and its vector position at time t is given by ~r(t) = (1 − t)
i +
1 − t
j.
(a) Find the Cartesian equation for the path of the particle.
(b) Assuming that P starts moving at time t = 0, how far will it have traveled at t = 1/2 seconds?
- A cannon sits atop a cliff that is 176 feet tall. It is aimed at an angle of 53. 1301
◦ above the horizontal,
and it shoots a projectile at the speed of 25 feet per second.
(a) Find the initial velocity vector, ~v(0) of the projectile.
(b) Using part(a), the fact that ~a(t) = − 32
j, and assuming ~r(0) = 176
j, find the velocity, ~v(t), and the
position, ~r(t), of the projectile for any time t.
(c) What is the maximum height of the projectile?
(d) At what time does the projectile hit the ground?
(e) With what speed does the projectile come smashing into the basin below the cliff?
- Let ~r(t) = t~i + t
j +
t
k.
(a) Find
T (1).
(b) Find
N (1).
(c) Find the curvature, κ at t = 1.
(d) Find the binormal
B(1).
