Q. What is the cross product of u = (1, 2, 3) and v = (4, 5, 6)?
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A.
(-3, 6, -3)
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B.
(0, 0, 0)
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C.
(3, -6, 3)
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D.
(1, 2, 3)
Solution
u × v = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3)
Correct Answer: A — (-3, 6, -3)
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Q. What is the cross product of vectors (1, 2, 3) and (4, 5, 6)?
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A.
(-3, 6, -3)
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B.
(0, 0, 0)
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C.
(3, -6, 3)
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D.
(1, 2, 3)
Solution
Cross product = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3)
Correct Answer: A — (-3, 6, -3)
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Q. What is the cross product of vectors A = (1, 2, 3) and B = (4, 5, 6)?
-
A.
(-3, 6, -3)
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B.
(0, 0, 0)
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C.
(3, -6, 3)
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D.
(1, -2, 1)
Solution
Cross product A × B = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3).
Correct Answer: A — (-3, 6, -3)
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Q. What is the distance between the points P(1, 2, 3) and Q(4, 5, 6)?
Solution
Distance = √((4-1)² + (5-2)² + (6-3)²) = √(9 + 9 + 9) = 3√3.
Correct Answer: A — 3√3
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Q. What is the dot product of vectors (1, 2) and (3, 4)?
Solution
Dot product = 1*3 + 2*4 = 3 + 8 = 11
Correct Answer: A — 11
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Q. What is the dot product of vectors A = (2, 3, 4) and B = (1, 0, -1)?
Solution
A . B = 2*1 + 3*0 + 4*(-1) = 2 + 0 - 4 = -2.
Correct Answer: B — 5
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Q. What is the equation of the line passing through the points (1, 2, 3) and (4, 5, 6)?
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A.
x = 1 + 3t, y = 2 + 3t, z = 3 + 3t
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B.
x = 1 + t, y = 2 + t, z = 3 + t
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C.
x = 1 + t, y = 2 + 2t, z = 3 + 3t
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D.
x = 1 + 3t, y = 2 + 2t, z = 3 + t
Solution
Direction ratios = (3, 3, 3), hence the line equation is x = 1 + 3t, y = 2 + 3t, z = 3 + 3t.
Correct Answer: A — x = 1 + 3t, y = 2 + 3t, z = 3 + 3t
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Q. What is the magnitude of the vector (2, -3, 6)?
Solution
Magnitude = √(2^2 + (-3)^2 + 6^2) = √(4 + 9 + 36) = √49 = 7.
Correct Answer: B — 9
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Q. What is the magnitude of the vector (3, 4)?
Solution
Magnitude = √(3^2 + 4^2) = √(9 + 16) = √25 = 5
Correct Answer: A — 5
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Q. What is the magnitude of the vector C = (6, 8, 10)?
Solution
Magnitude |C| = √(6^2 + 8^2 + 10^2) = √(36 + 64 + 100) = √200 = 10√2.
Correct Answer: C — 14
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Q. What is the magnitude of the vector v = (3, -4)?
Solution
Magnitude of v = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5.
Correct Answer: A — 5
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Q. What is the projection of vector a = (3, 4) onto vector b = (1, 0)?
Solution
Projection = (a · b / |b|^2) * b = (3*1 + 4*0) / 1^2 * (1, 0) = 3 * (1, 0) = (3, 0).
Correct Answer: A — 3
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Q. What is the projection of vector A = (3, 4) onto vector B = (1, 2)?
Solution
Projection = (A . B / |B|^2) * B = (3*1 + 4*2) / (1^2 + 2^2) * (1, 2) = 11/5 * (1, 2) = (2.2, 4.4).
Correct Answer: B — 3
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Q. What is the resultant of the vectors (2, 3) and (-1, 4)?
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A.
(1, 7)
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B.
(3, 1)
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C.
(1, 1)
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D.
(2, 4)
Solution
Resultant = (2 + (-1), 3 + 4) = (1, 7).
Correct Answer: A — (1, 7)
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Q. What is the resultant of vectors (1, 2) and (-1, -2)?
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A.
(0, 0)
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B.
(1, 2)
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C.
(2, 4)
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D.
(1, 1)
Solution
Resultant = (1 - 1, 2 - 2) = (0, 0)
Correct Answer: A — (0, 0)
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Q. What is the scalar projection of vector (3, 4) onto (1, 0)?
Solution
Scalar projection = (3*1 + 4*0) / √(1^2) = 3
Correct Answer: A — 3
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Q. What is 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. What is the scalar triple product of the vectors (1, 2, 3), (4, 5, 6), and (7, 8, 9)?
Solution
Scalar triple product = (1, 2, 3) · ((4, 5, 6) × (7, 8, 9)) = 0.
Correct Answer: A — 0
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Q. What is the scalar triple product of vectors A = (1, 0, 0), B = (0, 1, 0), C = (0, 0, 1)?
Solution
Scalar triple product = A · (B × C) = 1.
Correct Answer: A — 1
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Q. What is the scalar triple product of vectors a = (1, 2, 3), b = (4, 5, 6), c = (7, 8, 9)?
Solution
Scalar triple product = a · (b × c). Since b and c are linearly dependent, b × c = 0, hence the scalar triple product is 0.
Correct Answer: A — 0
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Q. What is the unit vector in the direction of (3, 4)?
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A.
(3/5, 4/5)
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B.
(4/5, 3/5)
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C.
(1, 1)
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D.
(0, 0)
Solution
Unit vector = (3, 4) / √(3^2 + 4^2) = (3/5, 4/5).
Correct Answer: A — (3/5, 4/5)
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Q. What is the unit vector in the direction of (3, 4, 0)?
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A.
(0.6, 0.8, 0)
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B.
(0.8, 0.6, 0)
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C.
(1, 0, 0)
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D.
(0, 1, 0)
Solution
Unit vector = (3, 4, 0) / √(3^2 + 4^2) = (3, 4, 0) / 5 = (0.6, 0.8, 0).
Correct Answer: A — (0.6, 0.8, 0)
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Q. What is the unit vector in the direction of (4, 3)?
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A.
(4/5, 3/5)
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B.
(3/4, 4/3)
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C.
(1, 1)
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D.
(0, 1)
Solution
Unit vector = (4/5, 3/5) where magnitude = √(4^2 + 3^2) = 5
Correct Answer: A — (4/5, 3/5)
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Q. What is 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 = (4/5, 3/5) since magnitude = 5.
Correct Answer: A — (4/5, 3/5)
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Q. What is the unit vector in the direction of v = (3, 4)?
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A.
(0.6, 0.8)
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B.
(1, 1)
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C.
(3, 4)
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D.
(0, 0)
Solution
Unit vector = v / |v| = (3, 4) / 5 = (0.6, 0.8)
Correct Answer: A — (0.6, 0.8)
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Q. What is the unit vector in the direction of vector A = (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, 0)
Solution
Unit vector = A / |A| = (3, 4) / 5 = (0.6, 0.8).
Correct Answer: A — (0.6, 0.8)
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