Vector & 3D Geometry
Q. If A = 2i - j and B = -i + 3j, what is the value of A · B?
Solution
A · B = (2)(-1) + (-1)(3) = -2 - 3 = -5.
Correct Answer: A — -1
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Q. If A = 6i + 8j and B = 3i + 4j, what is the scalar product A · B?
Solution
A · B = (6)(3) + (8)(4) = 18 + 32 = 50.
Correct Answer: B — 54
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Q. If A = 7i + 24j and B = 24i - 7j, calculate A · B.
Solution
A · B = (7)(24) + (24)(-7) = 168 - 168 = 0.
Correct Answer: A — 0
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Q. If A = i + 2j + 3k and B = 4i + 5j + 6k, find 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(4, 5, 6) are two points in space, what is the vector AB?
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A.
(3, 3, 3)
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B.
(2, 3, 4)
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C.
(1, 1, 1)
-
D.
(0, 0, 0)
Solution
Vector AB = B - A = (4-1, 5-2, 6-3) = (3, 3, 3).
Correct Answer: A — (3, 3, 3)
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Q. If A(1, 2, 3) and B(4, 5, 6) are two points, what is the vector AB?
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A.
(3, 3, 3)
-
B.
(3, 3, 0)
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C.
(0, 0, 0)
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D.
(1, 1, 1)
Solution
Vector AB = B - A = (4-1, 5-2, 6-3) = (3, 3, 3).
Correct Answer: A — (3, 3, 3)
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Q. If A(2, 3, 4) and B(1, 0, -1) are two points in space, find the vector AB.
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A.
(1, 3, 5)
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B.
(1, -3, -5)
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C.
(1, 3, -5)
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D.
(1, -3, 5)
Solution
AB = B - A = (1 - 2, 0 - 3, -1 - 4) = (-1, -3, -5) = (1, 3, 5) in the opposite direction.
Correct Answer: A — (1, 3, 5)
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Q. If C = (0, 1, 2) and D = (3, 4, 5), what is C · D?
Solution
C · D = 0*3 + 1*4 + 2*5 = 0 + 4 + 10 = 14.
Correct Answer: A — 10
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Q. If C = (3, -1, 2) and D = (0, 4, -1), what is the value of C · D?
Solution
C · D = 3*0 + (-1)*4 + 2*(-1) = 0 - 4 - 2 = -6.
Correct Answer: A — -2
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Q. If C = (3, -1, 2) and D = (0, 4, -3), what is the value of C · D?
Solution
C · D = 3*0 + (-1)*4 + 2*(-3) = 0 - 4 - 6 = -10.
Correct Answer: A — -10
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Q. If C = (3, -1, 2) and D = (4, 2, 1), calculate C · D.
Solution
C · D = 3*4 + (-1)*2 + 2*1 = 12 - 2 + 2 = 12.
Correct Answer: B — 11
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Q. If C = (3, -2, 1) and D = (0, 4, -3), what is the value of C · D?
Solution
C · D = 3*0 + (-2)*4 + 1*(-3) = 0 - 8 - 3 = -11.
Correct Answer: A — -10
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Q. If E = (1, 1, 1) and F = (1, 2, 3), find E · F.
Solution
E · F = 1*1 + 1*2 + 1*3 = 1 + 2 + 3 = 6.
Correct Answer: D — 6
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Q. If E = (1, 1, 1) and F = (2, 2, 2), what is the scalar product E · F?
Solution
E · F = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.
Correct Answer: B — 6
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Q. If E = (2, -1, 3) and F = (1, 2, 0), what is E · F?
Solution
E · F = 2*1 + (-1)*2 + 3*0 = 2 - 2 + 0 = 0.
Correct Answer: A — -1
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Q. If E = (5, 5, 5) and F = (1, 2, 3), what is the scalar product E · F?
Solution
E · F = 5*1 + 5*2 + 5*3 = 5 + 10 + 15 = 30.
Correct Answer: A — 30
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Q. If E = (x, y, z) and F = (2, 3, 4) such that E · F = 10, what is the equation relating x, y, z?
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A.
2x + 3y + 4z = 10
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B.
x + y + z = 10
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C.
x + 2y + 3z = 10
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D.
2x + 3y + z = 10
Solution
E · F = x*2 + y*3 + z*4 = 10 gives the equation 2x + 3y + 4z = 10.
Correct Answer: A — 2x + 3y + 4z = 10
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Q. If G = (1, 0, 1) and H = (0, 1, 0), find G · H.
Solution
G · H = 1*0 + 0*1 + 1*0 = 0 + 0 + 0 = 0.
Correct Answer: A — 0
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Q. If G = (1, 1, 1) and H = (1, -1, 1), what is G · H?
Solution
G · H = 1*1 + 1*(-1) + 1*1 = 1 - 1 + 1 = 1.
Correct Answer: A — 0
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Q. If G = (1, 1, 1) and H = (2, 2, 2), what is G · H?
Solution
G · H = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.
Correct Answer: A — 3
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Q. If G = (2, 2) and H = (3, -1), what is G · H?
Solution
G · H = 2*3 + 2*(-1) = 6 - 2 = 4.
Correct Answer: C — 5
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Q. If I = (1, 1, 1) and J = (2, 2, 2), what is the scalar product I · J?
Solution
I · J = 1*2 + 1*2 + 1*2 = 2 + 2 + 2 = 6.
Correct Answer: D — 6
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Q. If I = (1, 2, 3) and J = (3, 2, 1), what is I · J?
Solution
I · J = 1*3 + 2*2 + 3*1 = 3 + 4 + 3 = 10.
Correct Answer: A — 10
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Q. If I = (1, 2, 3) and J = (4, 5, 6), calculate I · J.
Solution
I · J = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Correct Answer: B — 30
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Q. If I = (a, b, c) and J = (2, 2, 2) such that I · J = 12, what is the relationship between a, b, c?
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A.
a + b + c = 6
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B.
2a + 2b + 2c = 12
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C.
a + b + c = 12
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D.
2a + 2b + 2c = 6
Solution
I · J = 2a + 2b + 2c = 12 gives the equation 2a + 2b + 2c = 12.
Correct Answer: B — 2a + 2b + 2c = 12
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Q. If M = (1, 2, 3) and N = (4, 5, 6), what is the scalar product M · N?
Solution
M · N = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Correct Answer: A — 32
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Q. If M = (2, 2, 2) and N = (3, 3, 3), what is M · N?
Solution
M · N = 2*3 + 2*3 + 2*3 = 6 + 6 + 6 = 18.
Correct Answer: A — 12
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Q. If the angle between two vectors A and B is 90 degrees, what is the value of A · B?
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A.
1
-
B.
0
-
C.
undefined
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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 position vector of a point is (5, 12), what is its distance from the origin?
Solution
Distance = √(5^2 + 12^2) = √(25 + 144) = √169 = 13
Correct Answer: A — 13
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