Vectors in Space

Maths, HSSC-Punjab First Year. This chapter is split into 49 topics and has 1,240 practice questions. The topics covered are listed below, followed by 10 free sample questions from the chapter.

Topics in this chapter

1. Scalars and vectors (recall) 22 questions
2. Volume of a tetrahedron 28 questions
3. Rectangular coordinate system in space 23 questions
4. Volume of a parallelepiped 26 questions
5. Distance between two points in space 30 questions
6. Vector joining two points and position vectors 25 questions
7. Projection of one vector along another 25 questions
8. Identity cos2 + cos2 + cos2 = 1 and direction angles 24 questions
9. Moment of a force (torque) 25 questions
10. Properties of the cross product 23 questions
11. Computing the scalar triple product 30 questions
12. Vectors in R3 and the basic operations 21 questions
13. Properties of vector addition and scalar multiplication 24 questions
14. Magnitudes, sums and angles between vectors 26 questions
15. Area of a parallelogram using the cross product 28 questions
16. Angle between vectors with standard angles 26 questions
17. Cross (vector) product: definition and direction 22 questions
18. Work done by a constant force 30 questions
19. Height of a parallelepiped 21 questions
20. Dot product of vectors in component form 30 questions
21. Parallel and perpendicular vectors: constants and classification 25 questions
22. Orthogonality (perpendicular vectors) 26 questions
23. Unit vector perpendicular to two vectors and the sine of the angle 29 questions
24. Cross products of i, j, k and parallel vectors 24 questions
25. Proofs and identities with the cross product 30 questions
26. Real-world applications of vector operations 22 questions
27. Direction angles and direction cosines (concepts) 21 questions
28. Vector perpendicular to two vectors (verification) 24 questions
29. Dot (scalar) product: definitions and meaning 22 questions
30. Dot products of i, j, k and properties 25 questions
31. Parallel vectors in space 22 questions
32. Vector proofs of triangle laws and trigonometric identities (dot product) 30 questions
33. Right-angled triangles by dot product 22 questions
34. Computing the cross product of two vectors 30 questions
35. Area of a triangle using the cross product 27 questions
36. Linear combinations of vectors and their magnitudes 28 questions
37. Direction cosines of given vectors 27 questions
38. Angle between two vectors (cosine formula) 30 questions
39. Scalar triple product: definition and determinant formula 22 questions
40. Unit vectors i, j, k and components of a vector 27 questions
41. Vectors forming a triangle and parallel sides 23 questions
42. Coplanar vectors and coplanar points 26 questions
43. Magnitude of a vector in space 30 questions
44. Vector of given magnitude parallel to a given vector 22 questions
45. Proofs of the properties of vector addition 21 questions
46. Determinant form of the cross product 25 questions
47. Unit vector of a given vector 24 questions
48. Properties and identities of the scalar triple product 20 questions
49. Vectors in the plane (recall of R2 operations) 27 questions

Sample questions: Vectors in Space

10 sample questions with the correct answer and an explanation. Try to answer each one before reading the explanation.

Q1. Which of the following is a vector quantity?

• A. Volume
• B. Momentum(correct answer)
• C. Temperature
• D. Density

Answer: B.A vector has both magnitude and direction; the book lists momentum among its vector examples, while temperature, volume and density are scalars.

Q2. A scalar is a quantity that has

• A. only direction, such as the heading of a ship
• B. magnitude and direction, written as a directed line
• C. only magnitude or size, such as mass or time(correct answer)
• D. both magnitude and direction, such as force or weight

Answer: C.The book defines a scalar as a quantity with magnitude (size) only, for example mass, time, density, temperature, length, volume, speed and work.

Q3. Work is classified in the book as

• A. a scalar, although force and displacement are vectors(correct answer)
• B. neither a scalar nor a vector, since it has units
• C. a vector, because work has a direction of motion
• D. a vector, because force and displacement are vectors

Answer: A.Work appears in the list of scalars: it is a scalar quantity obtained from two vectors (later as the dot product of force and displacement).

Q4. Geometrically, a vector is represented by

• A. a closed circle with a radius
• B. a directed line segment AB(correct answer)
• C. a point on a number line
• D. a pair of parallel lines

Answer: B.A vector is drawn as a directed line segment AB with an initial point and a terminal point; the arrow shows the direction and the length shows the magnitude.

Q5. The zero vector, by the convention in the book, is

• A. parallel to every vector(correct answer)
• B. perpendicular to no vector
• C. parallel only to itself
• D. parallel to no vector

Answer: A.Since O has no specific direction, we adopt the convention that the zero vector is parallel (and, as its dot product is zero, also perpendicular) to every vector.

Q6. The property (u + v) + w = u + (v + w) is called the

• A. Associative property(correct answer)
• B. Scalar multiplication property
• C. Distributive property
• D. Inverse for vector addition

Answer: A.In the book's list of properties, (u + v) + w = u + (v + w) is the associative property: the grouping of three vectors in a sum does not matter.

Q7. The property u + (−1)u = u − u = o is called the

• A. Inverse for vector addition(correct answer)
• B. Distributive property
• C. Additive identity
• D. Scalar multiplication property

Answer: A.In the book's list of properties, u + (−1)u = u − u = o is the inverse for vector addition: a vector plus its negative gives the zero vector.

Q8. Which triple can be the direction angles of a vector?

• A. 60°, 60°, 60°
• B. 30°, 60°, 90°(correct answer)
• C. 30°, 30°, 60°
• D. 45°, 60°, 90°

Answer: B.3/4 + 1/4 + 0 = 1. The others fail: (30, 30, 60) gives 3/4 + 3/4 + 1/4 = 7/4; (45, 60, 90) gives 1/2 + 1/4 + 0 = 3/4; (60, 60, 60) gives 3/4.

Q9. The result of the dot product of two vectors is

• A. a scalar (a real number)(correct answer)
• B. a vector parallel to the larger one
• C. a vector perpendicular to both
• D. a unit vector

Answer: A.The book stresses that the dot product is a scalar number, while the cross product is a vector quantity; it is also called the scalar or inner product.

Q10. For a = 2i + 3j − k and b = i − 2j + 4k (Exercise 14.2, Q4 i), the scalar projection of a along b, (a · b)/|b|, is

• A. −8/√14
• B. −8/√21(correct answer)
• C. 8/√21
• D. −8/21

Answer: B.a · b = 2 − 6 − 4 = −8 and |b| = √(1 + 4 + 16) = √21, so the projection of a along b is −8/√21. (The book's key lists vector multiples instead.)