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)?
-
A.
(1, 7)
-
B.
(3, 1)
-
C.
(1, 1)
-
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)?
-
A.
(0, 0)
-
B.
(1, 2)
-
C.
(2, 4)
-
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 product of A = (3, 4, 0) and B = (0, 0, 5)?
Solution
A · B = 3*0 + 4*0 + 0*5 = 0.
Correct Answer: A — 0
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Q. What is the scalar product of the unit vectors i and j?
Solution
i · j = 0, since they are orthogonal.
Correct Answer: B — 0
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Q. What is the scalar product of the vectors (3, 4) and (4, 3)?
Solution
The scalar product is 3*4 + 4*3 = 12 + 12 = 24.
Correct Answer: B — 25
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Q. What is the scalar product of the vectors (4, -3, 2) and (1, 1, 1)?
Solution
Scalar product = 4*1 + (-3)*1 + 2*1 = 4 - 3 + 2 = 3.
Correct Answer: D — 6
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Q. What is the scalar product of the vectors (5, -3) and (-2, 4)?
Solution
Scalar product = 5*(-2) + (-3)*4 = -10 - 12 = -22.
Correct Answer: A — -6
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Q. What is the scalar product of the vectors (5, 5, 5) and (1, 2, 3)?
Solution
Scalar product = 5*1 + 5*2 + 5*3 = 5 + 10 + 15 = 30.
Correct Answer: A — 30
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Q. What is the scalar product of the vectors A = (0, 1, 0) and B = (1, 0, 1)?
Solution
A · B = 0*1 + 1*0 + 0*1 = 0.
Correct Answer: A — 0
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Q. What is the scalar product of the vectors A = (1, 1, 1) and B = (1, 1, 1)?
Solution
A · B = 1*1 + 1*1 + 1*1 = 1 + 1 + 1 = 3.
Correct Answer: C — 3
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Q. What is the scalar product of the vectors A = (1, 2, 3) and B = (4, 5, 6)?
Solution
A · B = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Correct Answer: B — 30
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Q. What is the scalar product of the vectors A = (2, -1, 3) and B = (0, 4, -2)?
Solution
A · B = 2*0 + (-1)*4 + 3*(-2) = 0 - 4 - 6 = -10.
Correct Answer: A — -10
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Q. What is the scalar product of the vectors A = (4, 0, -3) and B = (0, 5, 2)?
Solution
A · B = 4*0 + 0*5 + (-3)*2 = 0 - 6 = -6.
Correct Answer: B — 0
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Q. What is the scalar product of the vectors K = (0, 1, 0) and L = (1, 0, 1)?
Solution
K · L = 0*1 + 1*0 + 0*1 = 0 + 0 + 0 = 0.
Correct Answer: A — 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)?
-
A.
(3/5, 4/5)
-
B.
(4/5, 3/5)
-
C.
(1, 1)
-
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)?
-
A.
(0.6, 0.8, 0)
-
B.
(0.8, 0.6, 0)
-
C.
(1, 0, 0)
-
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)?
-
A.
(4/5, 3/5)
-
B.
(3/4, 4/3)
-
C.
(1, 1)
-
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)?
-
A.
(4/5, 3/5)
-
B.
(3/5, 4/5)
-
C.
(1, 0)
-
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)?
-
A.
(0.6, 0.8)
-
B.
(1, 1)
-
C.
(3, 4)
-
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)?
-
A.
(0.6, 0.8)
-
B.
(0.8, 0.6)
-
C.
(1, 1)
-
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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