Q. Find the angle between the vectors A = 2i + 2j and B = 2i - 2j. (2022)
-
A.
0 degrees
-
B.
45 degrees
-
C.
90 degrees
-
D.
180 degrees
Solution
cos(θ) = (A · B) / (|A||B|). A · B = 0, hence θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. Find the angle between the vectors A = i + j and B = 2i + 2j.
-
A.
0 degrees
-
B.
45 degrees
-
C.
90 degrees
-
D.
60 degrees
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 1*2 = 4; |A| = √2, |B| = 2√2. Thus, cos(θ) = 4 / (√2 * 2√2) = 1, θ = 0 degrees.
Correct Answer:
A
— 0 degrees
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Q. Find the angle between the vectors A = i + j and B = i - j.
-
A.
0 degrees
-
B.
45 degrees
-
C.
90 degrees
-
D.
135 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (1 - 1) / (√2 * √2) = 0, θ = 90 degrees.
Correct Answer:
C
— 90 degrees
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Q. Find the angle between the vectors A = i + j and B = j - i. (2022)
-
A.
90 degrees
-
B.
45 degrees
-
C.
60 degrees
-
D.
30 degrees
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 0, hence θ = 90 degrees.
Correct Answer:
A
— 90 degrees
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Q. Find the unit vector in the direction of vector A = 6i - 8j.
-
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) = √(36 + 64) = 10. Unit vector = (6/10)i + (-8/10)j = (3/5)i - (4/5)j.
Correct Answer:
A
— 3/5 i - 4/5 j
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Q. Find the unit vector in the direction of vector D = -3i + 4j.
-
A.
-0.6i + 0.8j
-
B.
0.6i - 0.8j
-
C.
0.8i + 0.6j
-
D.
-0.8i + 0.6j
Solution
Magnitude |D| = √((-3)^2 + 4^2) = √(9 + 16) = √25 = 5. Unit vector = D/|D| = (-3/5)i + (4/5)j = -0.6i + 0.8j.
Correct Answer:
A
— -0.6i + 0.8j
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Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the distance from the origin to point P?
Solution
Distance = |r| = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29.
Correct Answer:
B
— 6
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Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the x-coordinate of point P? (2020)
Solution
The x-coordinate of point P is the coefficient of i in the position vector, which is 2.
Correct Answer:
A
— 2
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Q. If the position vector of point P is given by r = 2i + 3j + 4k, what is the x-component of r?
Solution
The x-component of r is 2.
Correct Answer:
A
— 2
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Q. If the position vector of point P is given by r = 3i + 4j, what is the distance of point P from the origin?
Solution
Distance = |r| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Correct Answer:
A
— 5
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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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