Q. If vector A = 6i + 8j, what is the unit vector in the direction of A? (2023)
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A.
3/5 i + 4/5 j
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B.
6/10 i + 8/10 j
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C.
1/5 i + 2/5 j
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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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Q. If vector J = 5i + 12j, what is the angle between J and the positive x-axis?
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A.
30 degrees
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B.
45 degrees
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C.
60 degrees
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D.
90 degrees
Solution
tan(θ) = 12/5; θ = tan^(-1)(12/5) which is approximately 67.38 degrees, closest to 60 degrees.
Correct Answer:
C
— 60 degrees
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Q. If vector K = 2i + 2j and vector L = -i + 3j, what is the resultant vector K + L?
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A.
i + 5j
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B.
i + j
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C.
3i + 5j
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D.
3i + j
Solution
K + L = (2i + 2j) + (-i + 3j) = (2 - 1)i + (2 + 3)j = 1i + 5j = i + 5j.
Correct Answer:
A
— i + 5j
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Q. What is the angle between the vectors A = 2i + 2j and B = 2i - 2j?
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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.
180 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (8) / (√8 * √8) = 1, θ = 0 degrees.
Correct Answer:
C
— 90 degrees
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Q. What is the angle between the vectors A = i + j and B = 2i + 2j?
-
A.
45 degrees
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B.
60 degrees
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C.
90 degrees
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D.
30 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (1*2 + 1*2) / (√2 * √8) = 4 / (2√2) = √2. θ = 45 degrees.
Correct Answer:
A
— 45 degrees
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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 cross product of vectors A = i + 2j and B = 3i + 4j? (2021)
Solution
A × B = |i j k| |1 2 0| |3 4 0| = (0 - 0)i - (0 - 0)j + (4 - 6)k = -2k.
Correct Answer:
A
— -2k
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Q. What is the cross product of vectors E = i + 2j and F = 3i + 4j?
Solution
E × F = |i j k| |1 2 0| |3 4 0| = (0 - 0)i - (0 - 0)j + (1*4 - 2*3)k = -2k.
Correct Answer:
A
— -2k
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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 = 1i + 2j?
Solution
Projection = (A · B / |B|^2) * B = (11 / 5) * (1i + 2j) = 2.5.
Correct Answer:
A
— 2.5
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Q. What is the projection of vector A = 3i + 4j onto vector B = 2i + 0j?
Solution
Projection of A onto B = (A · B / |B|^2)B. A · B = 6, |B|^2 = 4. Projection = (6/4)(2i) = 3i.
Correct Answer:
A
— 6
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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. What is the projection of vector A = 6i + 8j onto vector B = 2i + 2j?
Solution
Projection of A onto B = (A · B / |B|^2) * B = ((6*2 + 8*2) / (2^2 + 2^2)) * (2i + 2j) = (28/8)(2i + 2j) = 7i + 7j.
Correct Answer:
A
— 8
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Q. What is the scalar projection of vector H = 6i + 8j onto vector I = 3i + 4j?
Solution
Scalar projection = (H · I) / |I| = (6*3 + 8*4) / √(3^2 + 4^2) = (18 + 32) / 5 = 50 / 5 = 10.
Correct Answer:
C
— 10
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Q. What is the scalar triple product of vectors A = i + j + k, B = 2i + 3j + k, and C = 3i + j + 2k?
Solution
Scalar triple product = A · (B × C) = |i j k| |1 1 1| |2 3 1| |3 1 2| = 0.
Correct Answer:
D
— 0
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Q. What is the unit vector in the direction of vector A = 4i + 3j?
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A.
(4/5)i + (3/5)j
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B.
(3/4)i + (4/3)j
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C.
(4/3)i + (3/4)j
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D.
(3/5)i + (4/5)j
Solution
Unit vector = A / |A| = (4i + 3j) / √(4^2 + 3^2) = (4i + 3j) / 5 = (4/5)i + (3/5)j.
Correct Answer:
A
— (4/5)i + (3/5)j
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Q. Which of the following vectors is orthogonal to the vector A = 2i + 3j?
-
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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