Q. If A = (1, 0) and B = (0, 1), what is the angle between them?
-
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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Q. If A = (1, 2, 3) and B = (4, 5, 6), what is the value of A · B?
Solution
A · B = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Correct Answer: B — 30
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Q. If A = (1, 2, 3) and B = (k, k, k) are perpendicular, what is the value of k?
Solution
A · B = 1*k + 2*k + 3*k = 6k = 0. Thus, k = 0.
Correct Answer: D — 0
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Q. If A = (1, 2, 3) and B = (x, y, z) are such that A · B = 0, what is the condition for x, y, z?
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A.
x + 2y + 3z = 0
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B.
x - 2y + 3z = 0
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C.
x + 2y - 3z = 0
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D.
x - 2y - 3z = 0
Solution
A · B = 1*x + 2*y + 3*z = 0 gives the condition x + 2y + 3z = 0.
Correct Answer: A — x + 2y + 3z = 0
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Q. If A = (1, 2, 3) and B = (x, y, z) such that A · B = 14, find the value of x + y + z.
Solution
A · B = 1*x + 2*y + 3*z = 14. If we assume x = 2, y = 4, z = 2, then x + y + z = 8.
Correct Answer: C — 7
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Q. If A = (2, 0, -1) and B = (0, 3, 4), what is A · B?
Solution
A · B = 2*0 + 0*3 + (-1)*4 = 0 - 4 = -4.
Correct Answer: B — 0
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Q. If A = (2, 0, -1) and B = (0, 3, 4), what is the scalar product A · B?
Solution
A · B = 2*0 + 0*3 + (-1)*4 = 0 - 4 = -4.
Correct Answer: B — 0
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Q. If A = (2, 3) and B = (4, 5), what is the vector AB?
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A.
(2, 2)
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B.
(2, 3)
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C.
(4, 5)
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D.
(6, 8)
Solution
AB = B - A = (4 - 2, 5 - 3) = (2, 2)
Correct Answer: A — (2, 2)
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Q. If A = (2, 3) and B = (4, 7), find the vector AB.
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A.
(2, 4)
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B.
(2, 3)
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C.
(2, 1)
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D.
(2, 2)
Solution
Vector AB = B - A = (4 - 2, 7 - 3) = (2, 4).
Correct Answer: A — (2, 4)
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Q. If A = (2, 3) and B = (k, 1) are such that A · B = 10, find k.
Solution
A · B = 2k + 3*1 = 10. Thus, 2k + 3 = 10, leading to 2k = 7, k = 3.5.
Correct Answer: C — 3
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Q. If A = (2, 3, 4) and B = (0, 0, 0), what is A · B?
Solution
A · B = 2*0 + 3*0 + 4*0 = 0.
Correct Answer: A — 0
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Q. If A = (2, 3, 4) and B = (1, 0, -1), find the scalar product A · B.
Solution
A · B = 2*1 + 3*0 + 4*(-1) = 2 + 0 - 4 = -2.
Correct Answer: A — -1
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Q. If A = (2, 3, 4) and B = (1, 0, -1), find the vector product A × B.
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A.
(3, 6, -3)
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B.
(3, 4, -3)
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C.
(3, -4, 6)
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D.
(3, -6, 4)
Solution
A × B = |i j k|\n|2 3 4|\n|1 0 -1| = (3, 6, -3)
Correct Answer: A — (3, 6, -3)
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Q. If A = (2, 3, 4) and B = (1, 0, -1), what is the scalar product A · B?
Solution
A · B = 2*1 + 3*0 + 4*(-1) = 2 + 0 - 4 = -2.
Correct Answer: D — 10
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Q. If A = (2, 3, 4) and B = (k, 0, -1) are perpendicular, find k.
Solution
A · B = 2k + 3*0 + 4*(-1) = 0. Thus, 2k - 4 = 0, k = 2.
Correct Answer: A — -4
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Q. If A = (2, 3, 4) and B = (x, y, z) such that A · B = 20, find the value of x + y + z.
Solution
A · B = 2x + 3y + 4z = 20. If we assume x = 2, y = 2, z = 2, then 2*2 + 3*2 + 4*2 = 20, thus x + y + z = 6.
Correct Answer: C — 7
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Q. If A = (3, -1, 2) and B = (k, 4, -1) are orthogonal, find k.
Solution
A · B = 3k - 4 - 2 = 0. Thus, 3k - 6 = 0, k = 2.
Correct Answer: A — -2
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Q. If A = (3, -2, 1) and B = (4, 0, -1), what is the value of A · B?
Solution
A · B = 3*4 + (-2)*0 + 1*(-1) = 12 + 0 - 1 = 11.
Correct Answer: A — -1
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Q. If A = (a, b, c) and B = (1, 2, 3) such that A · B = 0, what is the relation between a, b, and c?
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A.
a + 2b + 3c = 0
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B.
a - 2b + 3c = 0
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C.
a + b + c = 0
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D.
a - b - c = 0
Solution
A · B = a*1 + b*2 + c*3 = 0. Thus, a + 2b + 3c = 0.
Correct Answer: A — a + 2b + 3c = 0
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