Q. Find the angle between the vectors A = i + j and B = i - j.
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A.
0 degrees
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B.
45 degrees
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C.
90 degrees
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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)
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A.
90 degrees
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B.
45 degrees
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C.
60 degrees
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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. 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 = 3i + 4j, what is the angle between them? (2023)
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A.
0 degrees
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B.
90 degrees
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C.
45 degrees
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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 + 6j, are the vectors A and B parallel?
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A.
Yes
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B.
No
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C.
Cannot be determined
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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 = 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 = 4i + 3j and vector B = -3i + 4j, what is the angle between them?
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A.
90 degrees
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B.
45 degrees
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C.
60 degrees
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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)
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A.
3i + 5j
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B.
5i + j
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C.
3i + j
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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 = 5i + 12j and vector B = -5i + 12j, what is the resultant vector A + B? (2021)
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A.
0i + 24j
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B.
10i + 0j
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C.
0i + 12j
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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 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. What is the cross product of vectors A = i + 2j + 3k and B = 4i + 5j + 6k?
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A.
-3i + 6j - 3k
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B.
-3i + 6j + 3k
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C.
3i - 6j + 3k
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D.
3i + 6j - 3k
Solution
A × B = |i j k| |1 2 3| |4 5 6| = -3i + 6j - 3k.
Correct Answer: A — -3i + 6j - 3k
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Q. What is the magnitude of the vector C = 5i - 12j?
Solution
Magnitude |C| = √(5^2 + (-12)^2) = √(25 + 144) = √169 = 13.
Correct Answer: A — 13
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Q. What is the projection of vector A = 3i + 4j onto vector B = 2i + 2j?
Solution
Projection = (A · B / |B|^2) * B = (14 / 8) * (2i + 2j) = (7/4)(2i + 2j) = 3.5i + 3.5j.
Correct Answer: A — 5
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Q. Which of the following vectors is orthogonal to the vector A = 2i + 3j?
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A.
3i - 2j
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B.
-3i + 2j
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C.
2i + 3j
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D.
i + j
Solution
A vector is orthogonal if A · B = 0. For B = 3i - 2j, A · B = 6 - 6 = 0.
Correct Answer: A — 3i - 2j
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