Download Solutions - RIT

March 29, 2007
Name_________________________
Exam 1
Show all work. Partial credit may be given, but only if all work is shown. Point credit is
shown beside each problem for a total of 100 points. Give exact answers.
1. (15 points) Let u = <3, 1, -1>, v = <4, -2, 1>, and w = <4, 1, -2>. Find the following:
(a) v x w
i j
k
= 4 −2 1 = < 3,12,12 >
4 1 −2
(b) Cosine of the angle between u and v
cos θ =
u• v
=
| u || v |
9
=
11 21
9
231
(c) The vector projection of u onto w
projwu =
u• w
15
5
< 4,1,−2 > = < 4,1,−2 >
2 w=
|w|
21
7
2. (15 points) Find parametric equations for the line through the point (1, -4, 1) and
parallel to w in problem 1.
x = 1 + 4t
y = -4 + t
z = 1 - 2t
1
March 29, 2007
Name_________________________
3. (15 points) If 6x2 + y2 - 2z2 = 6, find the following:
(a) Give both the equation and the name (type of curve) of the trace of the surface in the
xz plane.
If y = 0, then 6x2 - 2z2 = 0
This is a hyperbola
(b) Give both the equation and name of the trace in the yz plane.
If x = 0,then y2 - 2z2 = 0
This is a hyperbola
(c) Give the name of the type of
surface.
Elliptic hyperboloid of one sheet
(d) Sketch the surface.
4. (15 points) Find the equation of the plane passing through the point (-1, 0, 1) and
containing the line x = 5t, y = 1 + t, z = -t.
Let the given point be Q and let P be the point on the line corresponding
to t = 0 so P = (0, 1, 0). Then The vector from P to Q is in the plane as well
as the vector in the direction of the line. Form the cross product of these
vectors to get a normal vector.
i
j k
n = −1 −1 1 = < 0,4,4 >
5 1 −1
The plane is 0(x + 1) + 4(y - 0) + 4(z - 1) = 0 or y + z = 1
5. Determine whether the following two lines are parallel, skew, or intersecting. If they
intersect, find the point of intersection.
x = 1 + 2t, y = 2 + 3t, z = 3 + 4t and x = -1 + 6s, y = 3 -s, z = -5 + 2s
The lines are not parallel because the direction numbers are not scalar multiples
of each other.
Solve: 1 + 2t = -1 + 6s and 2 + 3t = 3 - s simultaneously to get t = 1/5, s = 2/5.
2
March 29, 2007
Name_________________________
Substitute these values into 3 + 4t ?=? -5 + 2s to get 3 + 4(1/5) ?=? -5 + 2(2/5).
Since 19/5 is not equal to -21/5, the lines are skew.
6. (10 points) Find the parametric equations of the line of intersection of the planes
x + y + z = 1 and 2x - 3y + 4z = 5.
A vector in the direction of the line of intersection is
i
j k
n1 × n 2 = 1 1 1 =< 7,−2,−5 >
2 −3 4
To find a point on the line, set z = 0 and solve x + y = 1 and 2x - 3y = 5
simultaneously to get x = 8/5, y = -3/5. Thus, parametric equations of the line
are: x = 8/5 + 7t, y = -3/5 - 2t, z = 0 - 5t
7. (10 points) Find the distance between the planes 3x + y - 4z = 2 and 3x + y - 4z = 24.
Find a point in the first plane by letting y = z = 0 and x = 2/3. The distance from
this point to the second plane is:
3(2 / 3) + 0 + 0 − 24
2
2
2
3 + 1 + (−4)
=
22
26
8. (10 points) Find the volume of the parallelopiped with adjacent edges PQ, PR, and
PS where P(2, 0, -1), Q(4,1, 0), R(3, -1, 1) and S(2, -2, 2).
2 1 1
PQ • PR × PS = 1 −1 2 = −3 = 3 cubic units
0 −2 3
3