Exam
Name___________________________________
8) The plane through the
points P(-1, 3, -8) ,
Q(2, -6, 31) and R(1, 5, 2).
SHORT ANSWER. Write the word or phrase that best
completes each statement or answers the question.
Find the intersection.
1) -5x + 4y = 6, -6y + 9z =
-7
1)
2) -8x + 6y - 2z = 2,
-2x + 5y - 8z = 8
2)
3) x = -3 + 5t, y = 3 - 3t, z =
8 + 5t ; -9x + 9y - 3z = 3
3)
Use a calculator to find the acute angle between the
planes to the nearest thousandth of a radian.
9) 6x - 4y - 8z = 4 and- 2x + 9)
5y - 10z = -1
Calculate the requested distance.
10) The distance from the
point S(2, 9, -9) to the
plane 2x + 11y + 10z = 4
Find the volume of the parallelopiped spanned by u, v,
and w.
4) u =(1,1,1); v = (9, 3, 2);
4)
w =( 8 , 10, 5)
11) The distance from the
point S(-4, 6, 8) to the
line x = -10 + 11t, y = -5 +
10t, z = -7 + 2t
Find the length and direction (when defined) of u × v.
5)
5) u = 4i + 2j + 8k, v = -i - 2j
- 2k
10)
11)
Find the component form of the specified vector.
12) The unit vector that
12)
makes an angle -5π/3
with the positive x-axis
Find an equation for the line that passes through the
given point and satisfies the given conditions.
6) P = (-8, 7); perpendicular
6)
to v = -6i - 4j
Write the equation for the plane.
7) The plane through the
point A(-3, -5, 5)
perpendicular to the
vector from the origin to
A.
8)
Solve the problem.
13) For the vectors u and v
with magnitudes u = 3
and v = 6, find the angle
θ between u and v which
makes proju v = 1
7)
14) Find the area of the
triangle determined by
the points P(1, 1, 1), Q(-2,
-7, -1), and R(-7, -1, 4).
1
13)
14)
15) Find a unit vector
perpendicular to plane
PQR determined by the
points P(2, 1, 1), Q(1, 0, 0)
and R(2, 2, 2).
15)
Write one or more inequalities that describe the set of
points.
23) The slab bounded by the
23)
planes x = 3 and x = 8
(planes included)
16) The unit vectors u and v
are combined to produce
two new vectors a = u + v
and b = u - v. Show that a
and b are orthogonal.
Assume u ≠ v.
16)
Find the acute angle between the lines.
24) -3x - 7y = -5 and 5x - 3y
24)
= 2
Determine whether the following is always true or not
always true. Given reasons for your answers.
17) (u × v) · v = u · (u × v)
17)
18) (u × v) · v = 0
18)
19) u × v = -(v × u)
19)
20) (u × v) · w = u · (w × v)
20)
Describe the given set of points with a single equation or
with a pair of equations.
21) The circle in which the
21)
plane through the point (
10, -9, 8) perpendicular
to the x-axis meets the
sphere of radius 26
centered at the origin.
22) The set of points
equidistant from the
points (-2, 0, 0) and (11, 0,
0)
22)
2
Answer Key
Testname: PRACTICE TEST CHAPTER 12
1) x = 36t - 4
7
, y = 45t + , z = 30t
15
6
2) x = -38t + 3) - 15
19
, y = -60t + , z = -28t
7
14
42 60 277
, , 29 29 29
4) 32
5) 6 5; 2 5
5
i - k
5
5
-6x - 4y = 20
3x + 5y - 5z = -59
7x - 2y - z = -5
1.168 rad
3
10)
5
6)
7)
8)
9)
11)
12)
43,514
15
1
3
, 2 2
13) 70.53°
4773
14)
2
15)
1
(j - k)
2
16) u = ux i + uyj and v = vx i + vyj , so
a = u + v = (ux + vx) i + (uy + vy)j and b = u - v = (ux - vx) i + (uy - vy)j
Take the dot product a · b:
a · b = (u + v) · (u - v) = (ux + vx)(ux - vx) + (uy + vy)(uy - vy)
2
2
2
2
2
2
2
2
= u x - v x + u y - v y = ( u x + u y ) - ( v x + v y )
= u - v = 1 - 1 = 0
Since the dot product of the two non-zero vectors is zero they are orthogonol.
----17) ========================
Not always true; The statement is false if u ≠
v. This is a lie. They are always equal.
18) Always true because u × v and v are orthogonal
19) Always true by definition of the cross product
20) Not always true; (u × v) · w = u · (v × w), but v × w = -(w × v) from which it follows that the original equation false if
w × v ≠ 0.
21) x2 + y2 + z 2 = 676 and x = 10
22) x = 4.5
23) 3 ≤ x ≤ 8
24) 1.435 radians
3