Q. Calculate the scalar product of A = (1, 1, 1) and B = (2, 2, 2).
Solution
A · B = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.
Correct Answer: D — 6
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Q. Calculate the scalar product of the vectors (1, 0, 0) and (0, 1, 0).
Solution
Scalar product = 1*0 + 0*1 + 0*0 = 0.
Correct Answer: A — 0
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Q. Calculate the scalar product of the vectors (1, 2, 3) and (4, 5, 6).
Solution
Scalar product = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Correct Answer: A — 32
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Q. Calculate the scalar product of the vectors (2, 3, 4) and (4, 3, 2).
Solution
Scalar product = 2*4 + 3*3 + 4*2 = 8 + 9 + 8 = 25.
Correct Answer: A — 28
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Q. Calculate the scalar product of the vectors (3, 0, -3) and (1, 2, 1).
Solution
Scalar product = 3*1 + 0*2 + (-3)*1 = 3 + 0 - 3 = 0.
Correct Answer: A — 0
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Q. Calculate 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. Calculate the scalar product of the vectors A = (4, -1, 2) and B = (2, 3, 1).
Solution
A · B = 4*2 + (-1)*3 + 2*1 = 8 - 3 + 2 = 7.
Correct Answer: A — 10
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Q. Calculate the scalar product of the vectors K = (0, 1, 2) and L = (3, 4, 5).
Solution
K · L = 0*3 + 1*4 + 2*5 = 0 + 4 + 10 = 14.
Correct Answer: A — 10
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Q. Determine the scalar product of the vectors (0, 1, 2) and (3, 4, 5).
Solution
Scalar product = 0*3 + 1*4 + 2*5 = 0 + 4 + 10 = 14.
Correct Answer: B — 11
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Q. Determine the scalar product of the vectors A = (1, 1, 1) and B = (2, 2, 2).
Solution
A · B = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.
Correct Answer: C — 6
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Q. Determine the scalar product of the vectors A = (2, 2, 2) and B = (3, 3, 3).
Solution
A · B = 2*3 + 2*3 + 2*3 = 6 + 6 + 6 = 18.
Correct Answer: A — 12
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Q. Find the angle between the vectors A = (1, 2, 2) and B = (2, 0, 2).
-
A.
0°
-
B.
45°
-
C.
60°
-
D.
90°
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 2*0 + 2*2 = 6. |A| = √(1^2 + 2^2 + 2^2) = 3, |B| = √(2^2 + 0^2 + 2^2) = 2√2. cos(θ) = 6 / (3 * 2√2) = 1/√2, θ = 45°.
Correct Answer: C — 60°
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Q. Find the angle between the vectors A = (1, 2, 2) and B = (2, 1, 1).
-
A.
60°
-
B.
45°
-
C.
30°
-
D.
90°
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 2*1 + 2*1 = 6; |A| = √(1^2 + 2^2 + 2^2) = 3; |B| = √(2^2 + 1^2 + 1^2) = √6. Thus, cos(θ) = 6 / (3√6) = 1/√6, θ = 45°.
Correct Answer: B — 45°
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Q. Find the angle between the vectors A = (3, -2, 1) and B = (1, 1, 1) if A · B = |A||B|cos(θ).
-
A.
60°
-
B.
45°
-
C.
90°
-
D.
30°
Solution
A · B = 3*1 + (-2)*1 + 1*1 = 3 - 2 + 1 = 2. |A| = √(3^2 + (-2)^2 + 1^2) = √14, |B| = √3. cos(θ) = 2/(√14 * √3). θ = 60°.
Correct Answer: A — 60°
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Q. Find the angle between the vectors A = (3, -2, 1) and B = (1, 1, 1).
-
A.
60°
-
B.
45°
-
C.
90°
-
D.
30°
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 3*1 + (-2)*1 + 1*1 = 2. |A| = √(3^2 + (-2)^2 + 1^2) = √14, |B| = √3. θ = cos^(-1)(2/(√14 * √3)).
Correct Answer: A — 60°
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Q. Find the projection of vector A = (2, 3) onto vector B = (1, 1).
Solution
Projection of A onto B = (A · B) / |B|^2 * B; A · B = 2*1 + 3*1 = 5; |B|^2 = 1^2 + 1^2 = 2; Projection = (5/2)(1, 1) = (2.5, 2.5).
Correct Answer: A — 1
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Q. Find the projection of vector A = (3, 4) onto vector B = (1, 2).
Solution
Projection of A onto B = (A · B) / |B|^2 * B. A · B = 3*1 + 4*2 = 11, |B|^2 = 1^2 + 2^2 = 5. Thus, projection = (11/5) * (1, 2) = (11/5, 22/5).
Correct Answer: B — 2
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Q. Find the scalar product of 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. Find the scalar product of the vectors (3, -2, 5) and (1, 4, -1).
Solution
Scalar product = 3*1 + (-2)*4 + 5*(-1) = 3 - 8 - 5 = -10.
Correct Answer: A — -1
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Q. Find the scalar product of the vectors (4, 5) and (1, 2).
Solution
Scalar product = 4*1 + 5*2 = 4 + 10 = 14.
Correct Answer: A — 14
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Q. Find the scalar product of the vectors (7, 8, 9) and (0, 1, 2).
Solution
Scalar product = 7*0 + 8*1 + 9*2 = 0 + 8 + 18 = 26.
Correct Answer: A — 26
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Q. Find the scalar product of the vectors A = (2, 3) and B = (4, -1).
Solution
A · B = 2*4 + 3*(-1) = 8 - 3 = 5.
Correct Answer: C — 10
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Q. Find the scalar product of the vectors A = 5i + 12j and B = 3i - 4j.
Solution
A · B = (5)(3) + (12)(-4) = 15 - 48 = -33.
Correct Answer: A — -33
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Q. Find the scalar product of the vectors G = (2, -3, 1) and H = (4, 0, -2).
Solution
G · H = 2*4 + (-3)*0 + 1*(-2) = 8 + 0 - 2 = 6.
Correct Answer: A — -2
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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 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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