Quiz 1 Practice
Math 2400 Calculus 3
Name:
Note: Some of these questions may require a calculator.
1. If |~v | = 7, find the coordinates of ~v .
y
x
50◦
~v
2. Which pairs of the vectors
√
3~i + ~j, 3~i +
√
3~j, ~i −
√
3~j are parallel and which are perpendicular?
3. What values of a make ~v = h2a, −a, 16i orthogonal to w
~ = h5, a, −1i?
4. Write ~a = h3, 2, −6i as the sum of two vectors, one parallel, and one perpendicular to d~ = h2, −4, 1i.
5. Let ~v and w
~ be vectors in R3 . If |~v | = 12, |~v − w|
~ = 13, and ~v · w
~ = 0, what is the area of the parallelogram
determined by ~v and w?
~
6. Determine whether each statement below is true or false.
(a) For any vectors ~v and ~u in R3 and any scalar k, k(~u · ~v ) = (k~u) · ~v .
(b) The center of the circle given by x2 − 6x + y 2 − 2y + 14 + z 2 = 12z is contained in the first octant.
(c) For any vectors ~v and ~u in R3 , |~u||~v | = ~u · ~v
7. Suppose ~v · w
~ = 8 and ~v × w
~ = h12, −3, 4i. Let θ be the acute angle between ~v and w.
~
(a) Find tan θ
(b) Find θ
8. Given the points P (1, 2, 3), Q(3, 5, 7), and R(2, 5, 3), find:
(a) A unit vector perpendicular to a plane containing P , Q, and R
(b) The angle between P Q and P R
(c) The area of the triangle P QR
(d) A unit vector parallel to P R
(e) The projection of P Q onto P R
9. [Challenge!] Find a vector with all of following properties:
• Magnitude 10
• Angle of 45◦ with the positive x-axis
• Angle of 75◦ with the positive y-axis
• Positive ~k-component
1
Quiz 1 Practice
Math 2400 Calculus 3
Answers:
1. h7 cos 50, −7 sin 50i
√
√
√
2. 3~i + ~j and 3~i + 3~j are parallel, and ~i − 3~j is perpendicular to the other two
3. a = 2, 8
−16 32 −8
58 −116 29
4. ~a =
, ,
+
,
,
where the first vector is parallel to d~ and the second is perpendicular
21 21 21
21 21 21
5. 60 square units
6. True, True, False
7. 13/8, ≈ 58.39◦
−12 4 3
3
11 33
1
◦
√ , √ ,0 ,
8.
, ,
, ≈ 49.76 , 13/2,
, ,0
13 13 13
10 10
10
10
+
*
r
√
1
9. [Challenge] 5 2, 10 cos 70◦ , 10
− cos2 (75◦ )
2
2