Math 4A
Practice Midterm 1
1. Consider the line L(t) = ⟨2t + 1, t − 2, 3t + 1⟩ and the point P = (4, −1, 5)
(a) Find a line parallel to L containing P
(b) Find the distance between L(0) and P
(c) Find a plane containing L and P
2. Consider the plane 2x + 10y − 11z = 60
(a) Find a line that goes through the origin and is orthogonal to this plane.
(b) Find the distance from this plane to the origin.
(c) Find the line of intersection of this plane and 2x − 10y + 11z = −60
3. Consider the vectors ⃗u = ⟨6, −6, 7⟩ and ⃗v = ⟨2, −2, 1⟩
(a) Find angle between ⃗u and ⃗v
(b) Find a unit vector w
⃗ which is orthogonal to ⃗u and ⃗v
(c) Find the projection of ⃗u onto ⃗v
4. Determine what type of surface is being described and find its equation.
(a) The cross sections perpendicular to the z axis are all circles, the cross sections perpendicular
to the x and y axes are hyperbolas.
(b) The cross sections perpendicular to the x axis are all the same hyperbola.
(c) The curve y = x2 is rotated around the y axis.
(d) The collection of points which are all exactly 2 units away from (1, 0, −1)
√
5. Consider the space curve ⃗r(t) = t, t2 , −4 t for t ≥ 0
(a) Evaluate lim ⃗r(t)
t→4
(b) Find the length of this curve for 0 ≤ t ≤ 1
(c) Find a parameterization of the line tangent to ⃗r(t) when t = 4
6. Let A be the point (0, −2, −2).
(a) Write A in cylindrical coordinates.
(b) Write A in spherical coordinates.
(c) Find the area of the triangle with vertices A, (4, −1, 1), and (1, 2, 1).
7. Consider the surface (x2 + y 2 − 2x)2 = 4(x2 + y 2 )
(a) Sketch the intersection of this surface with the plane z = 0
(b) Parameterize the intersection of this surface with z = x2 + y 2
8. An asteroid is spotted at location (4, 0, −6) at time t = 2 as it travels through space with
velocity function ⃗v (t) = ⟨t cos(πt), t sin(πt), t⟩
(a) Find the speed of this asteroid at time t = 1
(b) Where did it begin at time t = 0?
(c) Find the plane containing the asteroid’s position, and the vectors representing the
instantaneous velocity and acceleration at time t = 2