Scalar and vector (cross and dot) products - One Line Questions
1.
If vector a = 3i - j + 2k and vector b = i + 2j - k, what is a x b? —
-3i - 5j + 8k
2.
If vector a = (2, -1, 3) and vector b = (1, 3, -2), what is a . b? —
-5
3.
If vector a = 2i + 3j - k and vector b = i - 2j + 2k, what is the dot product a . b? —
-5
4.
If vector a = (1, 2, 0) and vector b = (0, 3, 1), what is a x b? —
(2, -1, 3)
5.
Given a = i + 2j and b = 3i + 4j, what is a x b? —
(4 - 6)k = -2k
6.
The scalar triple product [a b c] is defined as: —
a . (b x c)
7.
Which of the following is true for any vectors a and b? —
(a . b)^2 + |a x b|^2 = |a|^2 |b|^2
8.
The dot product a . b can be expressed as: —
|a||b| cos(theta)
9.
If vector a = i + j and vector b = i - j, what is the value of a . b? —
0
10.
What is the value of i . j? —
0
11.
What is the value of i x j? —
k
12.
If |a| = 2, |b| = 5, and the angle between a and b is 90 degrees, what is |a x b|? —
10
13.
If vector a = i + j + k, what is a . a? —
3
14.
What is the value of i . (j x k)? —
1
15.
If vector a = (1, 0, 0) and vector b = (0, 1, 0), what is a x b? —
k
16.
If vector a = 2i + 3j + 4k and vector b = 2i + 3j + 4k, what is a x b? —
0
17.
If vector a = (1, 2, 3) and vector b = (4, 5, 6), what is the value of [a b a]? —
0
18.
If vector a = 2i + j and vector b = i - 3j, what is the angle between them? —
90 degrees
19.
What is the area of the triangle with vertices at the origin, A = (1, 2), and B = (3, 4)? —
1
20.
If vector a = (1, 1, 1) and vector b = (1, -1, 1), what is a . b? —
1
21.
If vector a = 2i - j + k and vector b = i + 2j - 3k, what is a . b? —
2 - 2 - 3 = -3
22.
If vector a = i + 2j + 3k and vector b = 2i + 3j + 4k, what is a . b? —
26
23.
If a . b = |a x b|, then the angle theta between vectors a and b is: —
45 degrees
24.
What is the projection of vector a onto vector b if a . b = 12 and |b| = 4? —
3
25.
Given a = i + 2j and b = 3i + 4j, what is a . b? —
3 + 8 = 11
26.
What is the area of the parallelogram formed by vectors a = 3i + j and b = i + 2j? —
7
27.
If |a| = 3, |b| = 4, and the angle between a and b is 60 degrees, what is a . b? —
6
28.
If vector a = 3i + 2j and vector b = 2j + 3k, what is the cross product a x b? —
6i - 9j + 6k
29.
If vector a = 2i and vector b = 3j, what is a x b? —
6k
30.
What is the scalar projection of vector a = 3i + 4j onto vector b = i + j? —
7/√2
31.
If a, b, and c are vectors, which of the following is incorrect? —
(a + b) x c = a x c + b x c
32.
Which identity is known as Lagrange's identity for vectors? —
|a x b|^2 = |a|^2 |b|^2 - (a . b)^2
33.
What is the magnitude of the cross product |a x b|? —
|a||b| sin(theta)
34.
What is the condition for two non-zero vectors a and b to be perpendicular? —
a . b = 0
35.
The dot product is commutative, meaning: —
a . b = b . a
36.
The cross product a x b is zero if: —
a and b are parallel or anti-parallel
37.
If a and b are vectors such that a . b = 0 and |a x b| = 0, what can be concluded? —
a or b is the zero vector
38.
Which property does the cross product NOT satisfy? —
Associativity
39.
The scalar triple product [a b c] represents the: —
Volume of the parallelepiped formed by a, b, and c
40.
If a . b = 0 and a x b = 0, then what must be true? —
At least one of a or b is the zero vector.
41.
Which property does the dot product NOT satisfy? —
Associativity
42.
What is the vector projection of vector a = i + j onto vector b = i? —
i
43.
What is the value of i x (j x k)? —
i
44.
What is the geometric interpretation of the dot product of two vectors? —
It is related to the cosine of the angle between the vectors.
45.
If a . b = 0, then the vectors a and b are: —
Perpendicular
46.
The cross product a x b is a vector that is: —
Perpendicular to the plane containing a and b
47.
What is the geometric meaning of the magnitude of the cross product |a x b|? —
The area of the parallelogram formed by a and b
48.
The direction of the vector product a x b is given by: —
The right-hand rule
49.
The cross product of a vector with itself is always: —
The zero vector
50.
If [a b c] = 0, what can be said about vectors a, b, and c? —
They are coplanar.