Vector & 3D Geometry
Q. If the position vector of a point is given by r = (2t, 3t, 4t), what is the velocity vector?
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A.
(2, 3, 4)
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B.
(4, 6, 8)
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C.
(2t, 3t, 4t)
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D.
(0, 0, 0)
Solution
Velocity vector = dr/dt = (2, 3, 4)
Correct Answer: A — (2, 3, 4)
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Q. If the position vector of a point P is (2, 3, 4), what is the distance from the origin to point P?
Solution
Distance = √(2^2 + 3^2 + 4^2) = √(4 + 9 + 16) = √29 ≈ 5.385.
Correct Answer: B — 6
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Q. If the position vector of a point P is (x, y, z) and the vector a = (1, 2, 3), what is the projection of P onto a?
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A.
(1, 2, 3)
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B.
(2, 4, 6)
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C.
(0, 0, 0)
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D.
(x, y, z)
Solution
Projection of P onto a = ((P · a) / |a|^2) * a.
Correct Answer: D — (x, y, z)
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Q. If the position vector of a point P is given by r = (2t, 3t, 4t), find the coordinates of P when t = 1.
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A.
(2, 3, 4)
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B.
(1, 1, 1)
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C.
(0, 0, 0)
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D.
(2, 4, 6)
Solution
Substituting t = 1, r = (2*1, 3*1, 4*1) = (2, 3, 4).
Correct Answer: A — (2, 3, 4)
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Q. If the position vector of point P is (3, -2) and Q is (1, 4), what is the vector PQ?
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A.
(-2, 6)
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B.
(2, -6)
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C.
(4, -6)
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D.
(6, 2)
Solution
Vector PQ = Q - P = (1 - 3, 4 - (-2)) = (-2, 6).
Correct Answer: A — (-2, 6)
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Q. If the position vector of point P is (3, 4) and Q is (1, 2), what is the vector PQ?
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A.
(2, 2)
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B.
(4, 6)
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C.
(2, 4)
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D.
(1, 1)
Solution
Vector PQ = Q - P = (1 - 3, 2 - 4) = (-2, -2).
Correct Answer: A — (2, 2)
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Q. If the scalar product of two vectors A and B is 0, what can be said about the vectors?
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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 vector A = (1, 2) and B = (2, 1), what is the angle between them?
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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
Cosine of angle = (A · B) / (|A| |B|) = (1*2 + 2*1) / (√5 * √5) = 4/5, angle = cos^(-1)(4/5).
Correct Answer: C — 45 degrees
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Q. If the vector a = (1, 2) and b = (3, 4), find the angle between them using the dot product.
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A.
0 degrees
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B.
90 degrees
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C.
45 degrees
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D.
60 degrees
Solution
cos(θ) = (a · b) / (|a| |b|). a · b = 1*3 + 2*4 = 11, |a| = √(1^2 + 2^2) = √5, |b| = √(3^2 + 4^2) = 5. Thus, cos(θ) = 11 / (√5 * 5) = 11 / (5√5), θ = 60 degrees.
Correct Answer: D — 60 degrees
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Q. If the vector a = (2, -1) and b = (1, 3), what is a + b?
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A.
(3, 2)
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B.
(1, 2)
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C.
(2, 2)
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D.
(3, 1)
Solution
a + b = (2 + 1, -1 + 3) = (3, 2)
Correct Answer: A — (3, 2)
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Q. If the vector a = (2, -1) and b = (1, 3), what is the cross product a × b?
Solution
Cross product in 2D = a1*b2 - a2*b1 = 2*3 - (-1)*1 = 6 + 1 = 7
Correct Answer: A — 5
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Q. If the vector a = (2, 2) and b = (2, -2), what is the angle between them?
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A.
90 degrees
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B.
45 degrees
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C.
0 degrees
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D.
180 degrees
Solution
Angle = cos⁻¹((a·b) / (|a||b|)) = cos⁻¹(0) = 90 degrees
Correct Answer: A — 90 degrees
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Q. If the vector a = (2, 2) is scaled by a factor of 3, what is the resulting vector?
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A.
(6, 6)
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B.
(3, 3)
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C.
(2, 2)
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D.
(1, 1)
Solution
Scaled vector = 3 * a = 3 * (2, 2) = (6, 6)
Correct Answer: A — (6, 6)
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Q. If the vector a = (2, 3) and b = (4, 1), what is the cross product a × b?
Solution
Cross product a × b = 2*1 - 3*4 = 2 - 12 = -10.
Correct Answer: A — -10
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Q. If the vector a = (2, 3) and b = (4, 1), what is the resultant vector a + b?
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A.
(6, 4)
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B.
(2, 4)
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C.
(4, 2)
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D.
(6, 2)
Solution
Resultant vector a + b = (2+4, 3+1) = (6, 4).
Correct Answer: A — (6, 4)
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Q. If the vector A = (2, 3) is multiplied by 2, what is the resulting vector?
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A.
(4, 6)
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B.
(2, 3)
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C.
(1, 1.5)
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D.
(0, 0)
Solution
Resulting vector = 2 * A = 2 * (2, 3) = (4, 6).
Correct Answer: A — (4, 6)
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Q. If the vector A = (2, 3) is reflected across the line y = x, what is the resulting vector?
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A.
(3, 2)
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B.
(2, 3)
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C.
(0, 0)
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D.
(1, 1)
Solution
Reflection across y = x gives vector (3, 2).
Correct Answer: A — (3, 2)
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Q. If the vector A = (2, 3) is scaled by a factor of 2, what is the resulting vector?
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A.
(4, 6)
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B.
(2, 3)
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C.
(1, 1.5)
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D.
(0, 0)
Solution
Scaled vector = 2 * A = 2 * (2, 3) = (4, 6).
Correct Answer: A — (4, 6)
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Q. If the vector a = (2, 3, 4) and b = (1, 0, -1), what is a + b?
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A.
(3, 3, 3)
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B.
(1, 3, 3)
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C.
(2, 3, 3)
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D.
(2, 3, 5)
Solution
a + b = (2+1, 3+0, 4-1) = (3, 3, 3).
Correct Answer: A — (3, 3, 3)
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Q. If the vector a = (2, 3, 4) and b = (1, 0, -1), what is the scalar triple product a · (b × a)?
Solution
The scalar triple product is 0 because a · (b × a) = 0.
Correct Answer: A — 0
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Q. If the vector a = (2, 3, 4) is scaled by a factor of 2, what is the resulting vector?
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A.
(4, 6, 8)
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B.
(2, 3, 4)
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C.
(1, 1.5, 2)
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D.
(0, 0, 0)
Solution
Scaling the vector a by 2 gives (2*2, 2*3, 2*4) = (4, 6, 8).
Correct Answer: A — (4, 6, 8)
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Q. If the vector a = (3, 4) and b = (1, 2), find the cross product a × b.
Solution
In 2D, a × b = a1*b2 - a2*b1 = 3*2 - 4*1 = 6 - 4 = 2.
Correct Answer: A — -2
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Q. If the vector a = (3, 4) is scaled by a factor of 2, what is the new vector?
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A.
(6, 8)
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B.
(3, 4)
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C.
(1.5, 2)
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D.
(0, 0)
Solution
New vector = 2 * (3, 4) = (6, 8).
Correct Answer: A — (6, 8)
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Q. If the vector a = (3, 4, 0) and b = (0, 0, 5), what is the magnitude of a × b?
Solution
Magnitude of a × b = |a||b|sin(90) = |(3, 4, 0)|| (0, 0, 5)| = 5√(3^2 + 4^2) = 15.
Correct Answer: A — 15
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Q. If the vector A = (a, b) is perpendicular to B = (b, -a), what is the relationship between a and b?
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A.
a = b
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B.
a = -b
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C.
a + b = 0
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D.
a - b = 0
Solution
A·B = ab - ab = 0, hence A and B are perpendicular if a = -b.
Correct Answer: B — a = -b
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Q. If the vectors A = (1, 2) and B = (2, 1) are given, what is the angle between them?
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A.
90 degrees
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B.
45 degrees
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C.
60 degrees
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D.
30 degrees
Solution
Cosine of angle θ = (A · B) / (|A| |B|) = (1*2 + 2*1) / (√5 * √5) = 4/5, θ = cos⁻¹(4/5).
Correct Answer: B — 45 degrees
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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 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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