Math 220
Sample Test 1C
ANSWERS
(not guaranteed correct)
1.
Find the equation of the set of all points equidistant from the points
A (–1,5,3) and B (6, 2, –2). Describe the set.
14x - 6y - 10z = 9 This is a plane, equidistant from each of A and B, which is perpendicular
to the line which joins A and B.
2.
Let A = 3i – 2j + 3k
a)
b)
3.
Find A  B
Find A  B
and B = 2i + 2j – 5k.
-13
4i + 21j + 10k
Find the vector equation of the line which contains the points (2,8,3) and (8,4,–1).
⟨x,y,z⟩ = ⟨2,8,3⟩ + t ⟨3,-2,-2⟩
4.
Find the approximate measure (to the nearest 10th of a degree) of the acute angle
between the planes x + y + z = 3 and x + 2y + 3z = 1.
22.2°
5.
Find the parametric equations for the line of intersection of the planes
x + y + z = 3 and x + 2y + 3z = 1.
x = 5+t
y = -2 -2t
z = t
6.
2
Sketch the graph of x
2
– y + 4z = 0 for 0 ≤ x ≤ 4, 0 ≤ y ≤ 16, 0 ≤ z ≤ 2
I don't have a scanner, so I can't scan a drawing of this for you, but. . . It is an elliptic
paraboloid with the axis of symmetry along the y-axis. It comes to a "point" (has a
"vertex") at the origin. Cross-sections in planes parallel to the xz-plane are ellipses
where the axis (of the ellipse) parallel to the x-axis is twice as long as the axis parallel to
2
the z-axis. The trace in the xy-plane is the parabola y = x and the trace in the yz-plane
2
is the parabola y = 4z .
7.
Consider the curve r(t) =
<cost, sint, sin2t>.
Find the vector equation for the line which
 2 2 
is tangent to this curve at the point  2 . 2 . 1 .


1
⟨x,y,z⟩ = 2 ⟨ 2 , 2 , 2⟩ + t ⟨-1,1,0⟩
8.
r(t) = (sint)i + (cost)j + (sint)k
Find:
9.
a)
T(0)
b)
N(0)
c)
(t)
1
2
⟨1, 0, 1⟩
⟨0, -1, 0⟩
2
(
2
1 + cos t
)
3
a(t) = <t, 0, cost>
v(0) = <0, 0, 0>
r(0) = <0, 0, 0>
Find r(t)
r(t) =
1
6
⟨t3, 0, 6 - 6cost⟩