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
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B.
10i + 0j
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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
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B.
6/10 i + 8/10 j
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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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Q. If vector J = 5i + 12j, what is the angle between J and the positive x-axis?
-
A.
30 degrees
-
B.
45 degrees
-
C.
60 degrees
-
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?
-
A.
i + 5j
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B.
i + j
-
C.
3i + 5j
-
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. If vectors A = 3i + 4j and B = 2i - j, what is the dot product A · B?
Solution
A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.
Correct Answer:
C
— 10
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Q. If vectors A and B are perpendicular, then A · B equals:
Solution
If A and B are perpendicular, then by definition A · B = |A||B|cos(90°) = 0.
Correct Answer:
A
— 0
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Q. The angle between the vectors A = i + j and B = j + k is:
-
A.
45°
-
B.
60°
-
C.
90°
-
D.
30°
Solution
cos(θ) = (A · B) / (|A||B|) = (0) / (√2 * √2) = 0, thus θ = 90°.
Correct Answer:
C
— 90°
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Q. The magnitude of the vector A = 4i - 3j + 12k is:
Solution
Magnitude |A| = √(4^2 + (-3)^2 + 12^2) = √(16 + 9 + 144) = √169 = 13.
Correct Answer:
B
— 14
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Q. The scalar product of two unit vectors is 0. What can be inferred about these vectors?
-
A.
They are parallel
-
B.
They are orthogonal
-
C.
They are collinear
-
D.
They are equal
Solution
If the scalar product is 0, the vectors are orthogonal.
Correct Answer:
B
— They are orthogonal
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Q. The scalar product of two unit vectors is 0.5. What is the angle between them?
-
A.
60°
-
B.
30°
-
C.
90°
-
D.
120°
Solution
cos(θ) = 0.5, θ = cos⁻¹(0.5) = 60°.
Correct Answer:
A
— 60°
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Q. The scalar product of two vectors A and B is 12, and the angle between them is 60°. If |A| = 4, find |B|.
Solution
A · B = |A||B|cos(θ) => 12 = 4|B|(0.5) => |B| = 6.
Correct Answer:
B
— 8
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Q. The unit vector in the direction of vector A = 6i - 8j is:
-
A.
3/5 i - 4/5 j
-
B.
6/10 i - 8/10 j
-
C.
1/5 i - 4/5 j
-
D.
2/5 i - 3/5 j
Solution
Unit vector = A/|A| = (6i - 8j)/√(6^2 + (-8)^2) = (6i - 8j)/10 = 3/5 i - 4/5 j.
Correct Answer:
A
— 3/5 i - 4/5 j
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Q. What is the angle between the vectors A = 2i + 2j and B = 2i - 2j?
-
A.
0 degrees
-
B.
45 degrees
-
C.
90 degrees
-
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
-
B.
60 degrees
-
C.
90 degrees
-
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?
-
A.
-3i + 6j - 3k
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B.
-3i + 6j + 3k
-
C.
3i - 6j + 3k
-
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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