1. What does a vector represent?
Answer:
Explanation:
By definition, a vector has both magnitude (how much) and direction (which way).
2. If two vectors are parallel, their cross product is:
Answer:
Explanation:
The cross product of parallel vectors is zero since they don't enclose any area.
3. The dot product of two perpendicular vectors is:
Answer:
Explanation:
Dot product of two perpendicular vectors is zero since cos(90°) is 0.
4. If |a| = 3 and |b| = 4, what is the maximum value of the dot product a.b?
Answer:
Explanation:
Maximum value of a.b is |a||b|cos(0°) = |a||b|.
5. A unit vector is a vector with magnitude:
Answer:
Explanation:
A unit vector has a magnitude of one.
6. Which of the following is not a scalar quantity?
Answer:
Explanation:
Velocity has both magnitude and direction, making it a vector.
7. If a and b are two vectors, then a x (b x a) is:
Answer:
Explanation:
Using vector triple product rule, a x (b x a) = (a.b)a – (a.a)b = -|a|^2b + (a.b)a. When a and b are orthogonal, the result is -a.
8. What does the cross product of two vectors result in?
Answer:
Explanation:
The cross product of two vectors gives another vector.
9. If the vectors a and b are collinear, then:
Answer:
Explanation:
For collinear vectors, their cross product is zero.
10. Which operation on two vectors produces a scalar?
Answer:
Explanation:
The dot product of two vectors results in a scalar.
11. If a vector has zero magnitude, it is called:
Answer:
Explanation:
A vector with no magnitude and arbitrary direction is a zero vector.
12. The vectors a, b, and c are said to be coplanar if:
Answer:
Explanation:
If the scalar triple product is zero, the vectors are coplanar.
13. If a.b = a.c and |b| = |c|, then:
Answer:
Explanation:
Given conditions imply b – c is perpendicular to a.
14. The angle between the vectors i and j is:
Answer:
Explanation:
i and j are perpendicular unit vectors.
15. If |a| = 5 and |b| = 7, and the angle between them is 60°, then a.b is:
Answer:
Explanation:
a.b = |a||b|cos(60°) = 5*7*0.5 = 21.
16. If the vectors a and b are orthogonal, their dot product is:
Answer:
Explanation:
Dot product of orthogonal vectors is zero since the angle between them is 90°.
17. The magnitude of the vector 3i – 4j is:
Answer:
Explanation:
Square root of (3^2 + (-4)^2) is 5.
18. The projection of a vector a on b is given by:
Answer:
Explanation:
The projection is the component of a in the direction of b.
19. If a + b = c, then b is:
Answer:
Explanation:
Rearranging the given equation, we get b = c – a.
20. The unit vector in the direction of the vector 4i + 3j is:
Answer:
Explanation:
To find the unit vector, divide each component by the magnitude of the vector.