Math 220-Multivariable Calculus
Name: ___________________
Exam#1-Practice
Page 1
Date: ______________________
Math 220: Multi-Variable Calculus
Exam #1 –Practice Problems
You can use following questions to familiarize yourself with material and format of the exam.
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1. Consider P = (1, 2 ) , Q (4,1) and R = ( 5,1) . Let u = PQ and v = PR .
A)
B)
C)
D)
E)
F)
G)
Write u an v in component form.
Write u as a liner combination of the standard unit vectors i and j.
Find the magnitude of v.
Find 2u + v .
Graph vector u in standard position and find the angle it makes with the positive x-axis.
Graph vectors u and v. Find the angle between the vectors.
Find proju v .
2. Find the component form of v given its magnitude is v = 8 and the angles it makes is θ= 60° with the positive xaxis.
3. Find the standard equation of the sphere that has endpoints ( 5, −2,3) and ( 0, 4, −3)
4. Complete the square to write the equation of the sphere in standard form. Find the center and radius.
x2 + y 2 + z 2 − 4 x − 6 y + 4 =
0
5. Use the vectors to determine whether the points P ( 3, 4, −1) , Q ( −1,6,9 ) , and R (5,3, −6) are collinear.
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6. Consider P = ( 5,0,0 ) , Q (4, 4,0) and R = ( 2,0,6 ) . Let u = PQ and v = PR . Find the dot product u v and find
the cross product u × v .
7. Let =
u
A)
B)
C)
D)
E)
3, −2,1 , v =
2, −4, −3 and w =
−1, 2, 2 .
Find the angle between u and v.
Determine the projection of w onto v.
Find the cross product u × v .
Find the area of the parallelogram with adjacent sides u and v.
Find the volume of the parallelepiped with adjacent sides u, v, and w.
8. Find the parametric and symmetric equations of the line through the points (3,0, 2) and (9,11,6) .
Math 220-Multivariable Calculus
Exam#1-Practice
Page 2
9. Find a set of parametric equations for the line that passes though the point (1, 2,3) and is perpendicular to the
xz − plane .
10. Find the equation of the plane that passes through P ( −3, −4, 2 ) , Q ( −3, 4,1) , and R (1,1, −2) .
ind the equation of the plane that contains the lines
x −1
x +1
= y= z + 1 and
= y −1 = z − 2 .
−2
−2
12. Find the distance between the point (1,0, 2 ) and the plane 2 x − 3 y + 6 z =
6.
13. Explain why the planes 5 x − 3 y + z =
2 and 5 x − 3 y + z =−3 are parallel? Then, find the distance between the
parallel planes.
14. Find the angle between the planes given by x − 2 y + z =
0 and 2 x + 3 y − 2 z =
0.
15. Consider the following questions.
A) What is the difference between a vector and a scalar? Explain.
B) How can you determine whether or not two vectors are orthogonal? Explain.
C) How can you determine whether or not two vectors are parallel? Explain.
D) How can you determine whether or not three vectors with a common initial point lie in the same plane? Explain.
Math 220-Multivariable Calculus
Exam#1-Practice
Answers:
1) A) u =3, −1 , v =4, −1
2)
B) u= 3i − j
C)
17
D) 10i − 3 j
Page 3
E)
F)
v = 4, 4 3
2
5
97

3)  x −  + ( y − 1) 2 + z 2 =
2
4

4)
( x − 2)
2
9
+ ( y − 3) 2 + z 2 =
5) Collinear
6) A) u = −1, 4,0 , v =
−3,0,6 B) Dot-product 3
64 48
7) A) 56.91° B) − 32
29 , 29 , 29
8)
9)
10)
11)
12)
C) 10i + 11j − 8k D)
285
Parametric : x =
3 + 6t , y =
11t , z =
2 + 4t ; Symmetric : ( x − 3) / 6 =
y /11 =−
( z 2) / 4
Parametric : x =
1, y =2 + t , z =3
27 x + 4 y + 32 z + 33 =
0
x + 2y =
1
87
13) 35 7
14) 53.55°
G)