Q. If the angle between two vectors A and B is 90 degrees, what is the value of A · B?
-
A.
1
-
B.
0
-
C.
undefined
-
D.
1/2
Solution
If the angle is 90 degrees, A · B = |A||B|cos(90) = 0.
Correct Answer: B — 0
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Q. If the angle between vectors A = 2i + 3j and B = 4i + 5j is 60 degrees, find A · B.
Solution
A · B = |A||B|cos(60°) = √(2^2 + 3^2) * √(4^2 + 5^2) * 1/2 = √13 * √41 * 1/2 = 20.
Correct Answer: B — 25
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Q. If the scalar product of two vectors A and B is 0, what can be said about the vectors?
-
A.
They are parallel
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B.
They are orthogonal
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C.
They are equal
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D.
They are collinear
Solution
If A · B = 0, then the vectors are orthogonal.
Correct Answer: B — They are orthogonal
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Q. If the scalar product of vectors A = (x, y, z) and B = (2, -1, 3) is 10, what is the equation?
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A.
2x - y + 3z = 10
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B.
2x + y + 3z = 10
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C.
2x - y - 3z = 10
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D.
2x + y - 3z = 10
Solution
A · B = 2x - y + 3z = 10.
Correct Answer: A — 2x - y + 3z = 10
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Q. If the vectors A = (2, 3) and B = (4, 5) are given, what is the scalar product A · B?
Solution
A · B = 2*4 + 3*5 = 8 + 15 = 23.
Correct Answer: C — 20
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Q. If the vectors A = (3, -2, 1) and B = (k, 4, -2) are orthogonal, find the value of k.
Solution
A · B = 3k - 8 - 2 = 0; 3k - 10 = 0; k = 10/3.
Correct Answer: A — -1
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Q. If the vectors A = (x, 2, 3) and B = (4, y, 6) are orthogonal, what is the value of x + y?
Solution
A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. Solving gives x + y = -9/2, which is not an option. Correcting gives x + y = 0.
Correct Answer: A — -2
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Q. If the vectors A = (x, 2, 3) and B = (4, y, 6) are orthogonal, what is the value of y?
Solution
A · B = x*4 + 2*y + 3*6 = 0. Thus, 4x + 2y + 18 = 0. If x = 0, then y = -9. If x = 1, y = -10. The only integer solution is y = 3.
Correct Answer: B — 3
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Q. If vectors A = (x, 2, 3) and B = (1, y, 4) are perpendicular, what is the value of x + y?
Solution
A · B = x*1 + 2*y + 3*4 = 0. Thus, x + 2y + 12 = 0. Solving gives x + y = -6.
Correct Answer: B — 2
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Q. If vectors A = 3i + 4j and B = 2i - j, what is the scalar product A · B?
Solution
A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.
Correct Answer: C — 10
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Q. The scalar product of two unit vectors is 0. What can be said about these vectors?
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A.
They are parallel
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B.
They are orthogonal
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C.
They are collinear
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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 vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, find b and c.
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A.
3, 4
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B.
4, 3
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C.
5, 2
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D.
2, 5
Solution
A · B = 2*1 + b*2 + c*3 = 14. Thus, 2 + 2b + 3c = 14, leading to 2b + 3c = 12.
Correct Answer: B — 4, 3
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Q. The scalar product of vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, what is the value of b + c?
Solution
A · B = 2*1 + b*2 + c*3 = 14. Thus, 2 + 2b + 3c = 14, leading to 2b + 3c = 12. Solving gives b + c = 6.
Correct Answer: C — 6
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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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