Vector & 3D Geometry
Q. Find the scalar product of the vectors G = (5, -3, 2) and H = (1, 1, 1).
Solution
G · H = 5*1 + (-3)*1 + 2*1 = 5 - 3 + 2 = 4.
Correct Answer: D — 3
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Q. Find the scalar projection of vector A = (3, 4) onto vector B = (1, 0).
Solution
Scalar projection = (A · B) / |B| = (3*1 + 4*0) / 1 = 3.
Correct Answer: A — 3
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Q. Find the scalar triple product of vectors A = (1, 2, 3), B = (4, 5, 6), and C = (7, 8, 9).
Solution
Scalar triple product = A · (B × C) = 0, as vectors are coplanar.
Correct Answer: A — 0
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Q. Find the unit vector in the direction of the vector (3, 4).
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A.
(0.6, 0.8)
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B.
(0.8, 0.6)
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C.
(1, 1)
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D.
(0.5, 0.5)
Solution
Magnitude = √(3^2 + 4^2) = 5. Unit vector = (3/5, 4/5) = (0.6, 0.8).
Correct Answer: A — (0.6, 0.8)
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Q. Find the unit vector in the direction of the vector (3, 4, 0).
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A.
(0.6, 0.8, 0)
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B.
(0.3, 0.4, 0)
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C.
(1, 1, 0)
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D.
(0, 0, 1)
Solution
Magnitude = √(3^2 + 4^2) = 5. Unit vector = (3/5, 4/5, 0) = (0.6, 0.8, 0).
Correct Answer: A — (0.6, 0.8, 0)
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Q. Find the unit vector in the direction of the vector (4, 3).
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A.
(4/5, 3/5)
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B.
(3/5, 4/5)
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C.
(1, 0)
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D.
(0, 1)
Solution
Unit vector = (4, 3) / √(4^2 + 3^2) = (4, 3) / 5 = (4/5, 3/5).
Correct Answer: A — (4/5, 3/5)
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Q. Find the unit vector in the direction of the vector (6, 8).
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A.
(0.6, 0.8)
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B.
(0.8, 0.6)
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C.
(1, 1)
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D.
(0.5, 0.5)
Solution
Magnitude = √(6^2 + 8^2) = √(36 + 64) = √100 = 10. Unit vector = (6/10, 8/10) = (0.6, 0.8).
Correct Answer: A — (0.6, 0.8)
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Q. Find the unit vector in the direction of the vector v = (4, -3).
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A.
(4/5, -3/5)
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B.
(3/5, 4/5)
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C.
(4/3, -3/4)
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D.
(3/4, 4/3)
Solution
Magnitude |v| = √(4^2 + (-3)^2) = √(16 + 9) = 5. Unit vector = (4/5, -3/5).
Correct Answer: A — (4/5, -3/5)
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Q. Find the value of k if the vectors A = (1, k, 2) and B = (2, 3, 4) are perpendicular.
Solution
A · B = 1*2 + k*3 + 2*4 = 0. Thus, 2 + 3k + 8 = 0, so 3k = -10, k = -10/3.
Correct Answer: A — 1
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Q. For the vectors A = (1, 0, 0) and B = (0, 1, 0), what is the scalar product A · B?
Solution
A · B = 1*0 + 0*1 + 0*0 = 0.
Correct Answer: A — 0
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Q. For vectors A = (2, 3) and B = (4, 5), find the scalar product A · B.
Solution
A · B = 2*4 + 3*5 = 8 + 15 = 23.
Correct Answer: A — 23
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Q. For vectors A = (3, -2, 1) and B = (1, 4, -2), find A · B.
Solution
A · B = 3*1 + (-2)*4 + 1*(-2) = 3 - 8 - 2 = -7.
Correct Answer: A — -1
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Q. Given A = 3i + 4j and B = 0i + 0j, find A · B.
Solution
A · B = (3)(0) + (4)(0) = 0.
Correct Answer: A — 0
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Q. Given vectors A = (2, -1, 3) and B = (4, 0, -2), find A × B.
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A.
(-1, -10, 4)
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B.
(1, 10, -4)
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C.
(10, -1, 4)
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D.
(10, 1, -4)
Solution
A × B = |i j k|\n|2 -1 3|\n|4 0 -2| = (-1, -10, 4)
Correct Answer: A — (-1, -10, 4)
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Q. Given vectors A = (x, y, z) and B = (1, 2, 3), if A · B = 14, what is the value of x + 2y + 3z?
Solution
A · B = x*1 + y*2 + z*3 = 14, thus x + 2y + 3z = 14.
Correct Answer: A — 14
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Q. Given vectors P = (4, 0, -3) and Q = (1, 2, 1), find the scalar product P · Q.
Solution
P · Q = 4*1 + 0*2 + (-3)*1 = 4 + 0 - 3 = 1.
Correct Answer: B — 5
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Q. Given vectors P = (4, 1, 0) and Q = (0, 2, 3), find the scalar product P · Q.
Solution
P · Q = 4*0 + 1*2 + 0*3 = 0 + 2 + 0 = 2.
Correct Answer: B — 6
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Q. Given vectors P = (4, 1, 0) and Q = (1, 2, 3), find the scalar product P · Q.
Solution
P · Q = 4*1 + 1*2 + 0*3 = 4 + 2 + 0 = 6.
Correct Answer: B — 11
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Q. If A = (1, 0) and B = (0, 1), what is the angle between them?
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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.
180 degrees
Solution
Angle = cos⁻¹((A·B) / (|A||B|)) = cos⁻¹(0) = 90 degrees
Correct Answer: B — 90 degrees
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Q. If A = (1, 0, -1) and B = (0, 1, 1), what is the scalar product A · B?
Solution
A · B = 1*0 + 0*1 + (-1)*1 = 0 + 0 - 1 = -1.
Correct Answer: A — 0
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Q. If A = (1, 0, 0) and B = (0, 1, 0), what is the value of A · B?
Solution
A · B = 1*0 + 0*1 + 0*0 = 0.
Correct Answer: A — 0
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Q. If A = (1, 0, 0) and B = (0, 1, 0), what is the vector product A × B?
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A.
(0, 0, 1)
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B.
(1, 0, 0)
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C.
(0, 1, 0)
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D.
(0, 0, 0)
Solution
A × B = (0, 0, 1) using the right-hand rule.
Correct Answer: A — (0, 0, 1)
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Q. If A = (1, 1, 1) and B = (1, 1, 1), what is the scalar product A · B?
Solution
A · B = 1*1 + 1*1 + 1*1 = 3.
Correct Answer: C — 3
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Q. If A = (1, 1, 1) and B = (2, 2, 2), what is A × B?
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A.
(0, 0, 0)
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B.
(1, 1, 1)
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C.
(2, 2, 2)
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D.
(3, 3, 3)
Solution
A × B = (0, 0, 0) since A and B are parallel.
Correct Answer: A — (0, 0, 0)
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Q. If A = (1, 1, 1) and B = (2, 2, 2), what is the scalar product A · B?
Solution
A · B = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.
Correct Answer: B — 6
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Q. If A = (1, 1, 1) and B = (x, y, z) such that A · B = 3, what is the value of x + y + z?
Solution
A · B = 1*x + 1*y + 1*z = x + y + z = 3.
Correct Answer: C — 3
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Q. If A = (1, 2) and B = (3, 4), what is the dot product A · B?
Solution
Dot product A · B = 1*3 + 2*4 = 3 + 8 = 11.
Correct Answer: A — 10
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Q. If A = (1, 2) and B = (3, 4), what is the midpoint M of AB?
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A.
(2, 3)
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B.
(1, 2)
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C.
(3, 4)
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D.
(4, 5)
Solution
Midpoint M = ((1+3)/2, (2+4)/2) = (2, 3).
Correct Answer: A — (2, 3)
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Q. If A = (1, 2, 3) and B = (0, 1, 0), what is the direction of the vector product A × B?
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A.
(2, -3, 1)
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B.
(3, 0, -1)
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C.
(1, 0, -1)
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D.
(1, 3, 0)
Solution
A × B = (2, -3, 1) gives direction (2, -3, 1).
Correct Answer: B — (3, 0, -1)
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Q. If A = (1, 2, 3) and B = (4, 5, 6), what is the magnitude of the vector product A × B?
Solution
Magnitude |A × B| = √(1^2 + 2^2 + 3^2) = √14.
Correct Answer: D — √14
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