Vector & 3D Geometry
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
Learn More →
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
Learn More →
Q. If u = (1, 2) and v = (3, 4), what is u + v?
-
A.
(4, 6)
-
B.
(2, 3)
-
C.
(1, 2)
-
D.
(3, 4)
Solution
u + v = (1 + 3, 2 + 4) = (4, 6)
Correct Answer: A — (4, 6)
Learn More →
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
Learn More →
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
Learn More →
Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is A + B?
-
A.
(5, 7, 9)
-
B.
(4, 5, 6)
-
C.
(1, 2, 3)
-
D.
(0, 0, 0)
Solution
A + B = (1+4, 2+5, 3+6) = (5, 7, 9).
Correct Answer: A — (5, 7, 9)
Learn More →
Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the angle between them?
-
A.
0 degrees
-
B.
30 degrees
-
C.
60 degrees
-
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
Learn More →
Q. If vector A = (1, 2, 3) and vector B = (4, 5, 6), what is the vector A - B?
-
A.
(-3, -3, -3)
-
B.
(3, 3, 3)
-
C.
(5, 7, 9)
-
D.
(0, 0, 0)
Solution
A - B = (1-4, 2-5, 3-6) = (-3, -3, -3).
Correct Answer: A — (-3, -3, -3)
Learn More →
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
Learn More →
Q. If vector A = (3, -2, 1) and vector B = (1, 4, -3), what is the cross product A × B?
-
A.
(-5, -10, 14)
-
B.
(5, 10, -14)
-
C.
(10, 14, 5)
-
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)
Learn More →
Q. If vectors A = (x, 2, 3) and B = (1, y, 4) are perpendicular, what is the value of x + y?
Solution
A · B = x*1 + 2*y + 3*4 = 0. Thus, x + 2y + 12 = 0. Solving gives x + y = -6.
Correct Answer: B — 2
Learn More →
Q. If vectors A = 3i + 4j and B = 2i - j, what is the scalar product A · B?
Solution
A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.
Correct Answer: C — 10
Learn More →
Q. The scalar product of two unit vectors is 0. What can be said about these vectors?
-
A.
They are parallel
-
B.
They are orthogonal
-
C.
They are collinear
-
D.
They are equal
Solution
If the scalar product is 0, the vectors are orthogonal.
Correct Answer: B — They are orthogonal
Learn More →
Q. The scalar product of vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, find b and c.
-
A.
3, 4
-
B.
4, 3
-
C.
5, 2
-
D.
2, 5
Solution
A · B = 2*1 + b*2 + c*3 = 14. Thus, 2 + 2b + 3c = 14, leading to 2b + 3c = 12.
Correct Answer: B — 4, 3
Learn More →
Q. The scalar product of vectors A = (a, b, c) and B = (1, 2, 3) is 14. If a = 2, what is the value of b + c?
Solution
A · B = 2*1 + b*2 + c*3 = 14. Thus, 2 + 2b + 3c = 14, leading to 2b + 3c = 12. Solving gives b + c = 6.
Correct Answer: C — 6
Learn More →
Q. What is the angle between the vectors (1, 0) and (0, 1)?
-
A.
0 degrees
-
B.
90 degrees
-
C.
45 degrees
-
D.
180 degrees
Solution
The angle between (1, 0) and (0, 1) is 90 degrees.
Correct Answer: B — 90 degrees
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
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
Learn More →
Q. What is the cross product of the vectors (1, 0, 0) and (0, 1, 0)?
-
A.
(0, 0, 1)
-
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)
Learn More →
Q. What is the cross product of the vectors (1, 2, 3) and (4, 5, 6)?
-
A.
(-3, 6, -3)
-
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)
Learn More →
Q. What is the cross product of u = (1, 2, 3) and v = (4, 5, 6)?
-
A.
(-3, 6, -3)
-
B.
(0, 0, 0)
-
C.
(3, -6, 3)
-
D.
(1, 2, 3)
Solution
u × v = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3)
Correct Answer: A — (-3, 6, -3)
Learn More →
Q. What is the cross product of vectors (1, 2, 3) and (4, 5, 6)?
-
A.
(-3, 6, -3)
-
B.
(0, 0, 0)
-
C.
(3, -6, 3)
-
D.
(1, 2, 3)
Solution
Cross product = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3)
Correct Answer: A — (-3, 6, -3)
Learn More →
Q. What is the cross product of vectors A = (1, 2, 3) and B = (4, 5, 6)?
-
A.
(-3, 6, -3)
-
B.
(0, 0, 0)
-
C.
(3, -6, 3)
-
D.
(1, -2, 1)
Solution
Cross product A × B = |i j k| |1 2 3| |4 5 6| = (-3, 6, -3).
Correct Answer: A — (-3, 6, -3)
Learn More →
Q. What is the distance between the points P(1, 2, 3) and Q(4, 5, 6)?
Solution
Distance = √((4-1)² + (5-2)² + (6-3)²) = √(9 + 9 + 9) = 3√3.
Correct Answer: A — 3√3
Learn More →
Q. What is the dot product of vectors (1, 2) and (3, 4)?
Solution
Dot product = 1*3 + 2*4 = 3 + 8 = 11
Correct Answer: A — 11
Learn More →
Q. What is the dot product of vectors A = (2, 3, 4) and B = (1, 0, -1)?
Solution
A . B = 2*1 + 3*0 + 4*(-1) = 2 + 0 - 4 = -2.
Correct Answer: B — 5
Learn More →
Q. What is the equation of the line passing through the points (1, 2, 3) and (4, 5, 6)?
-
A.
x = 1 + 3t, y = 2 + 3t, z = 3 + 3t
-
B.
x = 1 + t, y = 2 + t, z = 3 + t
-
C.
x = 1 + t, y = 2 + 2t, z = 3 + 3t
-
D.
x = 1 + 3t, y = 2 + 2t, z = 3 + t
Solution
Direction ratios = (3, 3, 3), hence the line equation is x = 1 + 3t, y = 2 + 3t, z = 3 + 3t.
Correct Answer: A — x = 1 + 3t, y = 2 + 3t, z = 3 + 3t
Learn More →
Q. What is the magnitude of the vector (2, -3, 6)?
Solution
Magnitude = √(2^2 + (-3)^2 + 6^2) = √(4 + 9 + 36) = √49 = 7.
Correct Answer: B — 9
Learn More →
Showing 151 to 180 of 210 (7 Pages)
