Q. If V is scheduled before W and X is scheduled after W, which of the following is correct? (2020)
-
A.
V is last
-
B.
W is last
-
C.
X is last
-
D.
V is first
Solution
X is last as it is scheduled after W, which is after V.
Correct Answer:
C
— X is last
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Q. If V is the wife of W and X is the son of W, what is V's relationship to X? (2023)
-
A.
Mother
-
B.
Sister
-
C.
Aunt
-
D.
Cousin
Solution
V is the mother of X as she is the wife of X's father, W.
Correct Answer:
A
— Mother
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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is A + B?
-
A.
(5, 7, 9)
-
B.
(4, 5, 6)
-
C.
(1, 2, 3)
-
D.
(0, 0, 0)
Solution
A + B = (1+4, 2+5, 3+6) = (5, 7, 9).
Correct Answer:
A
— (5, 7, 9)
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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the angle between them?
-
A.
0 degrees
-
B.
30 degrees
-
C.
60 degrees
-
D.
90 degrees
Solution
Cosine of angle θ = (A . B) / (|A| |B|) = (1*4 + 2*5 + 3*6) / (√14 * √77) = 0, hence θ = 90 degrees.
Correct Answer:
D
— 90 degrees
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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the vector A - B?
-
A.
(-3, -3, -3)
-
B.
(3, 3, 3)
-
C.
(5, 7, 9)
-
D.
(0, 0, 0)
Solution
A - B = (1-4, 2-5, 3-6) = (-3, -3, -3).
Correct Answer:
A
— (-3, -3, -3)
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Q. If vector A = (2, 2, 2) and vector B = (1, 1, 1), what is the scalar triple product A . (B × A)?
Solution
A . (B × A) = 0, since B × A = 0.
Correct Answer:
A
— 0
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Q. If vector A = (3, -2, 1) and vector B = (1, 4, -3), what is the cross product A × B?
-
A.
(-5, -10, 14)
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B.
(5, 10, -14)
-
C.
(10, 14, 5)
-
D.
(14, -5, 10)
Solution
A × B = |i j k|\n|3 -2 1|\n|1 4 -3| = (-5, -10, 14).
Correct Answer:
A
— (-5, -10, 14)
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Q. If vector A = 2i + 3j + k and vector B = i - 2j + 4k, what is the cross product A × B?
-
A.
-10i + 5j + 7k
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B.
10i - 5j - 7k
-
C.
10i + 5j + 7k
-
D.
-10i - 5j + 7k
Solution
A × B = |i j k|\n|2 3 1|\n|1 -2 4| = (3*4 - 1*(-2))i - (2*4 - 1*1)j + (2*(-2) - 3*1)k = 10i - 5j - 7k.
Correct Answer:
A
— -10i + 5j + 7k
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Q. If vector A = 2i + 3j and vector B = -i + 4j, what is the resultant vector R = A + B?
-
A.
i + 7j
-
B.
i + j
-
C.
3i + 7j
-
D.
3i + 4j
Solution
R = A + B = (2 - 1)i + (3 + 4)j = 1i + 7j.
Correct Answer:
A
— i + 7j
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Q. If vector A = 2i + 3j and vector B = 3i + 4j, what is the angle between them? (2023)
-
A.
0 degrees
-
B.
90 degrees
-
C.
45 degrees
-
D.
60 degrees
Solution
cos(θ) = (A · B) / (|A| |B|) = (2*3 + 3*4) / (√(2^2 + 3^2) * √(3^2 + 4^2)) = 0.6, θ = 53.13 degrees.
Correct Answer:
D
— 60 degrees
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Q. If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between A and B? (2023)
-
A.
0 degrees
-
B.
90 degrees
-
C.
45 degrees
-
D.
60 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (8 + 15) / (√(13) * √(41)). θ = 60 degrees.
Correct Answer:
D
— 60 degrees
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Q. If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between them?
-
A.
0 degrees
-
B.
90 degrees
-
C.
45 degrees
-
D.
60 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (2*4 + 3*5) / (√(2^2 + 3^2) * √(4^2 + 5^2)) = 0.5, θ = 60 degrees.
Correct Answer:
D
— 60 degrees
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Q. If vector A = 2i + 3j and vector B = 4i + 6j, are the vectors A and B parallel?
-
A.
Yes
-
B.
No
-
C.
Cannot be determined
-
D.
Only if scaled
Solution
Vectors A and B are parallel because B = 2A.
Correct Answer:
A
— Yes
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Q. If vector A = 2i + 3j and vector B = 4i + k, what is the angle between A and B?
-
A.
90 degrees
-
B.
60 degrees
-
C.
45 degrees
-
D.
30 degrees
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 8 + 0 + 3(0) = 8; |A| = √(2^2 + 3^2) = √13; |B| = √(4^2 + 1^2) = √17. θ = cos^(-1)(8/(√13 * √17)).
Correct Answer:
B
— 60 degrees
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Q. If vector A = 3i + 4j and vector B = 2i - j, what is the dot product A · B?
Solution
A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.
Correct Answer:
A
— 10
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Q. If vector A = 3i + 4j and vector B = 4i + 3j, what is the cross product A × B?
Solution
A × B = |i j k| |3 4 0| |4 3 0| = (0 - 0)i - (0 - 0)j + (9 - 12)k = -3k.
Correct Answer:
B
— 1k
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Q. If vector A = 4i + 3j and vector B = -3i + 4j, what is the angle between them?
-
A.
90 degrees
-
B.
45 degrees
-
C.
60 degrees
-
D.
30 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (4*-3 + 3*4) / (5*5) = 0, θ = 90 degrees.
Correct Answer:
A
— 90 degrees
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Q. If vector A = 4i + 3j and vector B = -i + 2j, what is the resultant vector A + B? (2019)
-
A.
3i + 5j
-
B.
5i + j
-
C.
3i + j
-
D.
5i + 5j
Solution
A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.
Correct Answer:
A
— 3i + 5j
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Q. If vector A = 4i + 3j and vector B = -i + 2j, what is the resultant vector R = A + B? (2019)
-
A.
3i + 5j
-
B.
5i + j
-
C.
3i + j
-
D.
5i + 5j
Solution
R = A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.
Correct Answer:
A
— 3i + 5j
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Q. If vector A = 4i + 3j and vector B = 3i + 4j, what is the angle between A and B?
-
A.
45 degrees
-
B.
60 degrees
-
C.
90 degrees
-
D.
135 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (12 + 12) / (5 * 5) = 24/25, θ = cos^(-1)(24/25) ≈ 60 degrees.
Correct Answer:
B
— 60 degrees
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Q. If vector A = 4i + 3j and vector B = 4i - 3j, what is the angle between A and B? (2019)
-
A.
0 degrees
-
B.
90 degrees
-
C.
180 degrees
-
D.
45 degrees
Solution
A · B = 16 - 9 = 7. |A| = 5, |B| = 5. cos(θ) = 7/(5*5) = 0.28, θ = 180 degrees.
Correct Answer:
C
— 180 degrees
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Q. If vector A = 5i + 12j and vector B = -5i + 12j, what is the angle between them?
-
A.
0 degrees
-
B.
90 degrees
-
C.
180 degrees
-
D.
45 degrees
Solution
A · B = (5)(-5) + (12)(12) = -25 + 144 = 119. Since A and B are in opposite directions, the angle is 180 degrees.
Correct Answer:
C
— 180 degrees
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Q. If vector A = 5i + 12j and vector B = -5i + 12j, what is the resultant vector A + B? (2021)
-
A.
0i + 24j
-
B.
10i + 0j
-
C.
0i + 12j
-
D.
5i + 12j
Solution
A + B = (5 - 5)i + (12 + 12)j = 0i + 24j.
Correct Answer:
A
— 0i + 24j
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Q. If vector A = 5i + 12j and vector B = 12i - 5j, what is the angle between A and B?
-
A.
30 degrees
-
B.
60 degrees
-
C.
90 degrees
-
D.
120 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (5*12 + 12*(-5)) / (√(169) * √(169)) = 0, θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. If vector A = 5i + 12j and vector B = 12i - 5j, what is the value of A × B?
Solution
A × B = |i j k| |5 12 0| |12 -5 0| = (0 - 0)i - (0 - 0)j + (5*-5 - 12*12)k = -85k.
Correct Answer:
A
— -85
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Q. If vector A = 5i + 12j and vector B = 5i - 12j, what is the angle between A and B?
-
A.
0 degrees
-
B.
90 degrees
-
C.
180 degrees
-
D.
45 degrees
Solution
A · B = 5*5 + 12*(-12) = 25 - 144 = -119. Since A · B < 0, angle is 180 degrees.
Correct Answer:
C
— 180 degrees
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Q. If vector A = 5i + 5j and vector B = 5i - 5j, what is the angle between A and B?
-
A.
0 degrees
-
B.
45 degrees
-
C.
90 degrees
-
D.
180 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (25 - 25) / (√(50) * √(50)) = 0, θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. If vector A = 5i + 5j and vector B = 5i - 5j, what is the angle between them?
-
A.
0 degrees
-
B.
45 degrees
-
C.
90 degrees
-
D.
135 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (25 - 25) / (5√2 * 5√2) = 0, θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. If vector A = 6i + 8j, what is the unit vector in the direction of A? (2023)
-
A.
3/5 i + 4/5 j
-
B.
6/10 i + 8/10 j
-
C.
1/5 i + 2/5 j
-
D.
2/5 i + 3/5 j
Solution
Magnitude |A| = √(6^2 + 8^2) = 10. Unit vector = (1/10)(6i + 8j) = (3/5)i + (4/5)j.
Correct Answer:
A
— 3/5 i + 4/5 j
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Q. If vector G = 4i - 3j + 2k, what is the y-component of G?
Solution
The y-component of G is -3.
Correct Answer:
B
— -3
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