Vector & 3D Geometry
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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Q. If A = (a, b, c) and B = (1, 2, 3), and A · B = 14, find a + b + c.
Solution
a*1 + b*2 + c*3 = 14. One possible solution is a = 2, b = 4, c = 0, so a + b + c = 6.
Correct Answer: C — 7
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Q. If A = (a, b, c) and B = (1, 2, 3), and A · B = 14, what is the equation?
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A.
a + 2b + 3c = 14
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B.
a - 2b + 3c = 14
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C.
a + 2b - 3c = 14
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D.
a - 2b - 3c = 14
Solution
A · B = a*1 + b*2 + c*3 = 14.
Correct Answer: A — a + 2b + 3c = 14
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Q. If A = (a, b, c) and B = (1, 2, 3), find the value of a if A · B = 10.
Solution
A · B = a*1 + b*2 + c*3 = 10. If b = 2 and c = 1, then a + 4 + 3 = 10, so a = 3.
Correct Answer: C — 3
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Q. If A = (a, b, c) and B = (1, 2, 3), what is the scalar product A · B?
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A.
a + 2b + 3c
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B.
a - 2b - 3c
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C.
a * b * c
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D.
a^2 + b^2 + c^2
Solution
A · B = a*1 + b*2 + c*3 = a + 2b + 3c.
Correct Answer: A — a + 2b + 3c
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Q. If A = (x, y) and B = (y, x), what is the scalar product A · B?
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A.
x^2 + y^2
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B.
xy
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C.
x^2 - y^2
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D.
0
Solution
A · B = x*y + y*x = 2xy.
Correct Answer: A — x^2 + y^2
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Q. If A = (x, y, z) and B = (1, 1, 1) such that A · B = 6, find x + y + z.
Solution
A · B = x + y + z = 6. Thus, x + y + z = 6.
Correct Answer: B — 6
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Q. If A = (x, y, z) and B = (1, 1, 1), find the scalar product A · B.
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A.
x + y + z
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B.
x - y + z
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C.
x + y - z
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D.
x - y - z
Solution
A · B = x*1 + y*1 + z*1 = x + y + z.
Correct Answer: A — x + y + z
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Q. If A = (x, y, z) and B = (1, 2, 3), and A · B = 14, find the value of x + y + z.
Solution
A · B = x*1 + y*2 + z*3 = 14; Let x + y + z = k; We can find values satisfying this equation.
Correct Answer: C — 7
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Q. If A = (x, y, z) and B = (2, 2, 2), and A · B = 12, what is the value of x + y + z?
Solution
A · B = 2x + 2y + 2z = 12, thus x + y + z = 6.
Correct Answer: A — 6
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Q. If A = (x, y, z) and B = (2, 3, 4), and A · B = 10, find the value of x + y + z.
Solution
x*2 + y*3 + z*4 = 10. One possible solution is x = 1, y = 1, z = 1, so x + y + z = 3.
Correct Answer: C — 3
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Q. If A = (x, y, z) and B = (2, 3, 4), find the value of x if A · B = 10.
Solution
A · B = x*2 + y*3 + z*4 = 10. If y = 0 and z = 0, then x = 5.
Correct Answer: B — 2
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Q. If A = 1i + 1j + 1k and B = 2i + 2j + 2k, what is A · B?
Solution
A · B = (1)(2) + (1)(2) + (1)(2) = 2 + 2 + 2 = 6.
Correct Answer: A — 6
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