JEE Main - Scalar and vector (cross and dot) products Online Test (MCQ Quiz)

1. If vector a = 2i + 3j - k and vector b = i - 2j + 2k, what is the dot product a . b?

A) -6 B) -5 C) 0 D) 5

2. What is the magnitude of the cross product of two vectors that are parallel to each other?

A) Zero B) One C) The product of their magnitudes D) The sum of their magnitudes

3. If vector a = 3i + 2j and vector b = 2j + 3k, what is the cross product a x b?

A) 6i - 9j + 6k B) -6i + 9j - 6k C) 6i + 9j + 6k D) -6i - 9j - 6k

4. What is the geometric interpretation of the dot product of two vectors?

A) It represents the area of the parallelogram formed by the vectors. B) It represents the volume of the parallelepiped formed by the vectors. C) It is related to the cosine of the angle between the vectors. D) It is related to the sine of the angle between the vectors.

5. If vector a = i + j and vector b = i - j, what is the value of a . b?

A) 0 B) 1 C) 2 D) -1

6. The cross product of a vector with itself is always:

A) The vector itself B) The negative of the vector C) The zero vector D) One

7. What is the condition for two non-zero vectors a and b to be perpendicular?

A) a . b = 1 B) a . b = |a||b| C) a . b = 0 D) a x b = 0

8. If vector a = 2i and vector b = 3j, what is a x b?

A) 6k B) -6k C) 0 D) 6

9. The dot product is commutative, meaning:

A) a . b = b . a B) a . b = - (b . a) C) a . b = |a| |b| D) a . b = 0

10. What is the magnitude of the cross product |a x b|?

A) a . b B) |a||b| cos(theta) C) |a||b| sin(theta) D) |a||b| tan(theta)

11. If vector a = i + 2j + 3k and vector b = 2i + 3j + 4k, what is a . b?

A) 20 B) 26 C) 30 D) 32

12. The cross product a x b is a vector that is:

A) Parallel to both a and b B) Perpendicular to both a and b C) Perpendicular to the plane containing a and b D) Perpendicular to the sum a + b

13. If vector a = 3i - j + 2k and vector b = i + 2j - k, what is a x b?

A) -3i - 5j + 8k B) 3i + 5j - 8k C) -3i + 5j - 8k D) 3i - 5j + 8k

14. What is the value of i . j?

A) 0 B) 1 C) i D) j

15. What is the value of i x j?

A) 0 B) 1 C) k D) -k

16. If |a| = 3, |b| = 4, and the angle between a and b is 60 degrees, what is a . b?

A) 6 B) 12 C) 3√3 D) 6√3

17. If |a| = 2, |b| = 5, and the angle between a and b is 90 degrees, what is |a x b|?

A) 0 B) 7 C) 10 D) 2.5

18. Which property does the dot product NOT satisfy?

A) Commutativity B) Distributivity over vector addition C) Associativity D) Scalar multiplication property

19. Which property does the cross product NOT satisfy?

A) Anti-commutativity (a x b = - (b x a)) B) Distributivity over vector addition C) Associativity D) Scalar multiplication property

20. If vector a = i + j + k, what is a . a?

A) 0 B) 1 C) 3 D) √3

21. The scalar triple product [a b c] is defined as:

A) (a . b) x c B) a . (b x c) C) (a x b) . c D) a x (b . c)

22. The scalar triple product [a b c] represents the:

A) Area of the parallelogram formed by a and b B) Volume of the parallelepiped formed by a, b, and c C) Length of vector a D) Dot product of a and b

23. If [a b c] = 0, what can be said about vectors a, b, and c?

A) They are mutually perpendicular. B) They are coplanar. C) They are parallel. D) Their magnitudes are equal.

24. What is the value of i . (j x k)?

A) 0 B) 1 C) -1 D) i

25. What is the value of i x (j x k)?

A) i B) j C) k D) 0

26. If vector a = (2, -1, 3) and vector b = (1, 3, -2), what is a . b?

A) -5 B) -4 C) -3 D) 0

27. If vector a = (1, 2, 0) and vector b = (0, 3, 1), what is a x b?

A) (2, -1, 3) B) (-2, 1, 3) C) (2, 1, 3) D) (-2, -1, 3)

28. The dot product a . b can be expressed as:

A) |a x b| B) |a||b| sin(theta) C) |a||b| cos(theta) D) |a||b| tan(theta)

29. The cross product a x b is zero if:

A) a and b are perpendicular B) a and b are parallel or anti-parallel C) a and b are orthogonal D) a . b = 0

30. If vector a = 2i + j and vector b = i - 3j, what is the angle between them?

A) 0 degrees B) 45 degrees C) 90 degrees D) 180 degrees

31. What is the area of the parallelogram formed by vectors a = 3i + j and b = i + 2j?

A) 5 B) 7 C) √59 D) √74

32. If a . b = 0 and a x b = 0, then what must be true?

A) At least one of a or b is the zero vector. B) a and b are parallel. C) a and b are perpendicular. D) a and b have the same magnitude.

33. Which of the following is true for any vectors a and b?

A) (a . b)^2 + |a x b|^2 = |a|^2 |b|^2 B) (a . b)^2 + |a x b|^2 = |a|^2 + |b|^2 C) (a . b)^2 + |a x b|^2 = |a|^2 - |b|^2 D) (a . b)^2 - |a x b|^2 = |a|^2 |b|^2

34. If vector a = (1, 1, 1) and vector b = (1, -1, 1), what is a . b?

A) 1 B) -1 C) 0 D) 2

35. If vector a = (1, 0, 0) and vector b = (0, 1, 0), what is a x b?

A) 0 B) i C) j D) k

36. The direction of the vector product a x b is given by:

A) The right-hand rule B) The left-hand rule C) The direction of a D) The direction of b

37. If vector a = 2i + 3j + 4k and vector b = 2i + 3j + 4k, what is a x b?

A) 0 B) 2(2i + 3j + 4k) C) 4(2i + 3j + 4k) D) i + j + k

38. What is the projection of vector a onto vector b if a . b = 12 and |b| = 4?

A) 3 B) 4 C) 12 D) 48

39. If a, b, and c are vectors, which of the following is incorrect?

A) a . (b + c) = a . b + a . c B) a x (b + c) = a x b + a x c C) (a + b) . c = a . c + b . c D) (a + b) x c = a x c + b x c

40. Given a = i + 2j and b = 3i + 4j, what is a . b?

A) 3 + 8 = 11 B) 4 + 6 = 10 C) 1 + 6 = 7 D) 3 + 2 = 5

41. Given a = i + 2j and b = 3i + 4j, what is a x b?

A) (4 - 6)k = -2k B) (3 - 8)k = -5k C) (1 - 2)k = -k D) (2 - 3)k = -k

42. What is the geometric meaning of the magnitude of the cross product |a x b|?

A) The area of the triangle formed by a and b B) The area of the rectangle formed by a and b C) The area of the parallelogram formed by a and b D) The volume of the parallelepiped formed by a and b

43. If a . b = |a x b|, then the angle theta between vectors a and b is:

A) 22.5 degrees B) 45 degrees C) 67.5 degrees D) 90 degrees

44. If vector a = (1, 2, 3) and vector b = (4, 5, 6), what is the value of [a b a]?

A) 0 B) 1 C) 3 D) 6

45. What is the scalar projection of vector a = 3i + 4j onto vector b = i + j?

A) 7/√2 B) 1/√2 C) 7 D) 1

46. If a and b are vectors such that a . b = 0 and |a x b| = 0, what can be concluded?

A) a is parallel to b B) a is perpendicular to b C) a or b is the zero vector D) a and b have the same magnitude

47. What is the vector projection of vector a = i + j onto vector b = i?

A) i B) j C) i + j D) 0

48. If vector a = 2i - j + k and vector b = i + 2j - 3k, what is a . b?

A) 2 - 2 + 3 = 3 B) 2 + 2 - 3 = 1 C) 2 - 2 - 3 = -3 D) 2 + 2 + 3 = 7

49. What is the area of the triangle with vertices at the origin, A = (1, 2), and B = (3, 4)?

A) 1 B) 2 C) 3 D) 4

50. If a . b = 0, then the vectors a and b are:

A) Parallel B) Perpendicular C) Coplanar D) Collinear

Scalar and vector (cross and dot) products - Online Test

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