Q. If vector A = 2i + 3j + k and vector B = i - 2j + 4k, what is the cross product A × B?
-
A.
-10i + 5j + 7k
-
B.
10i - 5j - 7k
-
C.
10i + 5j + 7k
-
D.
-10i - 5j + 7k
Solution
A × B = |i j k|\n|2 3 1|\n|1 -2 4| = (3*4 - 1*(-2))i - (2*4 - 1*1)j + (2*(-2) - 3*1)k = 10i - 5j - 7k.
Correct Answer: A — -10i + 5j + 7k
Learn More →
Q. If vector A = 2i + 3j and vector B = 3i + 4j, what is the angle between them? (2023)
-
A.
0 degrees
-
B.
90 degrees
-
C.
45 degrees
-
D.
60 degrees
Solution
cos(θ) = (A · B) / (|A| |B|) = (2*3 + 3*4) / (√(2^2 + 3^2) * √(3^2 + 4^2)) = 0.6, θ = 53.13 degrees.
Correct Answer: D — 60 degrees
Learn More →
Q. If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between A and B? (2023)
-
A.
0 degrees
-
B.
90 degrees
-
C.
45 degrees
-
D.
60 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (8 + 15) / (√(13) * √(41)). θ = 60 degrees.
Correct Answer: D — 60 degrees
Learn More →
Q. If vector A = 2i + 3j and vector B = 4i + 6j, are the vectors A and B parallel?
-
A.
Yes
-
B.
No
-
C.
Cannot be determined
-
D.
Only if scaled
Solution
Vectors A and B are parallel because B = 2A.
Correct Answer: A — Yes
Learn More →
Q. If vector A = 2i + 3j and vector B = 4i + k, what is the angle between A and B?
-
A.
90 degrees
-
B.
60 degrees
-
C.
45 degrees
-
D.
30 degrees
Solution
cos(θ) = (A · B) / (|A| |B|). A · B = 8 + 0 + 3(0) = 8; |A| = √(2^2 + 3^2) = √13; |B| = √(4^2 + 1^2) = √17. θ = cos^(-1)(8/(√13 * √17)).
Correct Answer: B — 60 degrees
Learn More →
Q. If vector A = 3i + 4j and vector B = 2i - j, what is the dot product A · B?
Solution
A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.
Correct Answer: A — 10
Learn More →
Q. If vector A = 3i + 4j and vector B = 4i + 3j, what is the cross product A × B?
Solution
A × B = |i j k| |3 4 0| |4 3 0| = (0 - 0)i - (0 - 0)j + (9 - 12)k = -3k.
Correct Answer: B — 1k
Learn More →
Q. If vector A = 4i + 3j and vector B = -3i + 4j, what is the angle between them?
-
A.
90 degrees
-
B.
45 degrees
-
C.
60 degrees
-
D.
30 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (4*-3 + 3*4) / (5*5) = 0, θ = 90 degrees.
Correct Answer: A — 90 degrees
Learn More →
Q. If vector A = 4i + 3j and vector B = -i + 2j, what is the resultant vector A + B? (2019)
-
A.
3i + 5j
-
B.
5i + j
-
C.
3i + j
-
D.
5i + 5j
Solution
A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.
Correct Answer: A — 3i + 5j
Learn More →
Q. If vector A = 4i + 3j and vector B = -i + 2j, what is the resultant vector R = A + B? (2019)
-
A.
3i + 5j
-
B.
5i + j
-
C.
3i + j
-
D.
5i + 5j
Solution
R = A + B = (4 - 1)i + (3 + 2)j = 3i + 5j.
Correct Answer: A — 3i + 5j
Learn More →
Q. If vector A = 4i + 3j and vector B = 4i - 3j, what is the angle between A and B? (2019)
-
A.
0 degrees
-
B.
90 degrees
-
C.
180 degrees
-
D.
45 degrees
Solution
A · B = 16 - 9 = 7. |A| = 5, |B| = 5. cos(θ) = 7/(5*5) = 0.28, θ = 180 degrees.
Correct Answer: C — 180 degrees
Learn More →
Q. If vector A = 5i + 12j and vector B = -5i + 12j, what is the resultant vector A + B? (2021)
-
A.
0i + 24j
-
B.
10i + 0j
-
C.
0i + 12j
-
D.
5i + 12j
Solution
A + B = (5 - 5)i + (12 + 12)j = 0i + 24j.
Correct Answer: A — 0i + 24j
Learn More →
Q. If vector A = 5i + 12j and vector B = 12i - 5j, what is the angle between A and B?
-
A.
30 degrees
-
B.
60 degrees
-
C.
90 degrees
-
D.
120 degrees
Solution
cos(θ) = (A · B) / (|A||B|) = (5*12 + 12*(-5)) / (√(169) * √(169)) = 0, θ = 90 degrees.
Correct Answer: C — 90 degrees
Learn More →
Q. If vector A = 5i + 12j and vector B = 12i - 5j, what is the value of A × B?
Solution
A × B = |i j k| |5 12 0| |12 -5 0| = (0 - 0)i - (0 - 0)j + (5*-5 - 12*12)k = -85k.
Correct Answer: A — -85
Learn More →
Q. If vector A = 6i + 8j, what is the unit vector in the direction of A? (2023)
-
A.
3/5 i + 4/5 j
-
B.
6/10 i + 8/10 j
-
C.
1/5 i + 2/5 j
-
D.
2/5 i + 3/5 j
Solution
Magnitude |A| = √(6^2 + 8^2) = 10. Unit vector = (1/10)(6i + 8j) = (3/5)i + (4/5)j.
Correct Answer: A — 3/5 i + 4/5 j
Learn More →
Q. If vector G = 4i - 3j + 2k, what is the y-component of G?
Solution
The y-component of G is -3.
Correct Answer: B — -3
Learn More →
Q. If vector J = 5i + 12j, what is the angle between J and the positive x-axis?
-
A.
30 degrees
-
B.
45 degrees
-
C.
60 degrees
-
D.
90 degrees
Solution
tan(θ) = 12/5; θ = tan^(-1)(12/5) which is approximately 67.38 degrees, closest to 60 degrees.
Correct Answer: C — 60 degrees
Learn More →
Q. If vector K = 2i + 2j and vector L = -i + 3j, what is the resultant vector K + L?
-
A.
i + 5j
-
B.
i + j
-
C.
3i + 5j
-
D.
3i + j
Solution
K + L = (2i + 2j) + (-i + 3j) = (2 - 1)i + (2 + 3)j = 1i + 5j = i + 5j.
Correct Answer: A — i + 5j
Learn More →
Q. If vectors A = 3i + 4j and B = 2i - j, what is the dot product A · B?
Solution
A · B = (3)(2) + (4)(-1) = 6 - 4 = 2.
Correct Answer: C — 10
Learn More →
Q. If vectors A and B are perpendicular, then A · B equals:
Solution
If A and B are perpendicular, then by definition A · B = |A||B|cos(90°) = 0.
Correct Answer: A — 0
Learn More →
Q. If x + y = 10 and xy = 21, what is the value of x^2 + y^2? (2021)
Solution
We know that x^2 + y^2 = (x + y)^2 - 2xy. Substituting the values, we get (10)^2 - 2(21) = 100 - 42 = 58.
Correct Answer: A — 49
Learn More →
Q. If x = -3, what is the value of |x| + x?
Solution
|-3| = 3, so |x| + x = 3 - 3 = 0.
Correct Answer: B — -3
Learn More →
Q. If x = -5, what is the value of |x|?
Solution
The absolute value |x| of -5 is 5.
Correct Answer: C — 5
Learn More →
Q. If x = 3 and y = 4, what is the value of x² + y²?
Solution
x² + y² = 3² + 4² = 9 + 16 = 25.
Correct Answer: A — 25
Learn More →
Q. If x = 4 and y = 5, what is the value of x² - y²?
Solution
x² - y² = 4² - 5² = 16 - 25 = -9.
Correct Answer: B — 9
Learn More →
Q. If x = 4, what is the value of 2x - 3? (2022)
Solution
2x - 3 = 2(4) - 3 = 8 - 3 = 5.
Correct Answer: C — 7
Learn More →
Q. If x = 4, what is the value of 3x - 2? (2019)
Solution
3x - 2 = 3(4) - 2 = 12 - 2 = 10.
Correct Answer: B — 12
Learn More →
Q. If x = 5, what is the value of (x - 1)²?
Solution
(x - 1)² = (5 - 1)² = 4² = 16.
Correct Answer: A — 16
Learn More →
Q. If x^2 - 5x + 6 = 0, what are the values of x? (2019)
-
A.
2, 3
-
B.
1, 6
-
C.
3, 2
-
D.
0, 6
Solution
Factoring gives (x - 2)(x - 3) = 0, so x = 2 or x = 3.
Correct Answer: A — 2, 3
Learn More →
Q. If z = 1 + i, find |z|².
Solution
|z|² = (1)² + (1)² = 1 + 1 = 2.
Correct Answer: B — 2
Learn More →
Showing 481 to 510 of 973 (33 Pages)
