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?
-
A.
(4, 6)
-
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?
-
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 u = (1, 2) and v = (3, 4), what is the dot product u · v?
Solution
Dot product u · v = 1*3 + 2*4 = 3 + 8 = 11.
Correct Answer: A — 10
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Q. If u = (1, 2) and v = (3, 4), what is u + v?
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A.
(4, 6)
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B.
(2, 3)
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C.
(1, 2)
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D.
(3, 4)
Solution
u + v = (1 + 3, 2 + 4) = (4, 6)
Correct Answer: A — (4, 6)
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Q. If u = (1, 2, 3) and v = (4, 5, 6), what is the dot product u · v?
Solution
Dot product u · v = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.
Correct Answer: B — 27
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Q. If u = (2, 3, 1) and v = (1, 0, -1), find the dot product u · v.
Solution
u · v = 2*1 + 3*0 + 1*(-1) = 2 + 0 - 1 = 1.
Correct Answer: A — 5
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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is A + B?
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A.
(5, 7, 9)
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B.
(4, 5, 6)
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C.
(1, 2, 3)
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D.
(0, 0, 0)
Solution
A + B = (1+4, 2+5, 3+6) = (5, 7, 9).
Correct Answer: A — (5, 7, 9)
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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the angle between them?
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A.
0 degrees
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B.
30 degrees
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C.
60 degrees
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D.
90 degrees
Solution
Cosine of angle θ = (A . B) / (|A| |B|) = (1*4 + 2*5 + 3*6) / (√14 * √77) = 0, hence θ = 90 degrees.
Correct Answer: D — 90 degrees
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Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the vector A - B?
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A.
(-3, -3, -3)
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B.
(3, 3, 3)
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C.
(5, 7, 9)
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D.
(0, 0, 0)
Solution
A - B = (1-4, 2-5, 3-6) = (-3, -3, -3).
Correct Answer: A — (-3, -3, -3)
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Q. If vector A = (2, 2, 2) and vector B = (1, 1, 1), what is the scalar triple product A . (B × A)?
Solution
A . (B × A) = 0, since B × A = 0.
Correct Answer: A — 0
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Q. If vector A = (3, -2, 1) and vector B = (1, 4, -3), what is the cross product A × B?
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A.
(-5, -10, 14)
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B.
(5, 10, -14)
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C.
(10, 14, 5)
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D.
(14, -5, 10)
Solution
A × B = |i j k|\n|3 -2 1|\n|1 4 -3| = (-5, -10, 14).
Correct Answer: A — (-5, -10, 14)
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Q. What is the angle between the vectors (1, 0) and (0, 1)?
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A.
0 degrees
-
B.
90 degrees
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C.
45 degrees
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D.
180 degrees
Solution
The angle between (1, 0) and (0, 1) is 90 degrees.
Correct Answer: B — 90 degrees
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Q. What is the angle between the vectors (1, 2, 2) and (2, 1, 2)?
-
A.
90 degrees
-
B.
60 degrees
-
C.
45 degrees
-
D.
30 degrees
Solution
Cosine of angle θ = (u · v) / (|u| |v|). Calculate to find θ = 60 degrees.
Correct Answer: B — 60 degrees
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Q. What is the angle between the vectors a = (1, 2, 2) and b = (2, 0, 2)?
-
A.
0 degrees
-
B.
45 degrees
-
C.
90 degrees
-
D.
60 degrees
Solution
cos(θ) = (a · b) / (|a| |b|). Calculate a · b = 1*2 + 2*0 + 2*2 = 6, |a| = √(1^2 + 2^2 + 2^2) = 3, |b| = √(2^2 + 0^2 + 2^2) = 2√2. Thus, cos(θ) = 6 / (3 * 2√2) = 1/√2, θ = 45 degrees.
Correct Answer: D — 60 degrees
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Q. What is the angle between the vectors u = (1, 0) and v = (0, 1)?
-
A.
0 degrees
-
B.
90 degrees
-
C.
45 degrees
-
D.
180 degrees
Solution
The angle between u and v is 90 degrees since they are perpendicular.
Correct Answer: B — 90 degrees
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Q. What is the angle between vectors A = (1, 0, 0) and B = (0, 1, 0)?
-
A.
0 degrees
-
B.
45 degrees
-
C.
90 degrees
-
D.
180 degrees
Solution
The angle θ = cos⁻¹((A . B) / (|A| |B|)) = cos⁻¹(0) = 90 degrees.
Correct Answer: C — 90 degrees
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Q. What is the cross product of the vectors (1, 0, 0) and (0, 1, 0)?
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A.
(0, 0, 1)
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B.
(1, 1, 0)
-
C.
(0, 0, 0)
-
D.
(1, 0, 0)
Solution
Cross product = (1, 0, 0) × (0, 1, 0) = (0, 0, 1).
Correct Answer: A — (0, 0, 1)
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Q. What is the cross product of the vectors (1, 2, 3) and (4, 5, 6)?
-
A.
(-3, 6, -3)
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B.
(-3, 6, 3)
-
C.
(3, -6, 3)
-
D.
(3, 6, -3)
Solution
Cross product = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3).
Correct Answer: A — (-3, 6, -3)
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Showing 31 to 60 of 86 (3 Pages)
