1st Year Math Chapter 14 Exercise 14.1 Notes: Introduction to Vectors in Space (Punjab Board 2026-27)
Complete concept notes, formulas, and solved examples for ICS/FSc Part 1 Mathematics, Punjab Board (PECTAA), Single National Curriculum 2026-27 session.
Chapter 14, Vectors in Space, closes the 14-unit Mathematics 11 (PECTAA) textbook, following Differentiation. Exercise 14.1 introduces vectors as quantities with both magnitude and direction, along with the basic vocabulary used throughout the chapter.
What Does This Exercise Cover?
This exercise builds the foundational vocabulary of vectors — magnitude, unit vectors, position vectors, and the conditions for vectors to be equal or parallel — before the dot and cross products are introduced.
Key Concepts
1. Scalar and Vector Quantities
A scalar is a quantity that has only magnitude (size), such as mass, time, or temperature. A vector is a quantity that has both magnitude and direction, such as displacement, velocity, or force.
2. Geometric Representation and Magnitude
Geometrically, a vector is represented by a directed line segment with an initial point and a terminal point. For a vector u = (x, y, z) in space, its magnitude is |u| = √(x² + y² + z²).
3. Unit Vector
A unit vector has magnitude 1 and points in the same direction as a given vector v. It’s found using û = v / |v|.
4. Null (Zero) Vector, Equal Vectors, and Parallel Vectors
A null vector has its initial and terminal points at the same location. Two vectors are equal if they have the same magnitude and direction. Two vectors are parallel if and only if one is a non-zero scalar multiple of the other.
5. Position Vector and Distance Between Two Points
A position vector describes the location of a point relative to the origin O; the position vector of point P is written OP. The distance between two points equals the magnitude of the vector connecting them.
Solved Examples
Example 1: Find the magnitude of the vector u = (3, 4, 12).
|u| = √(3² + 4² + 12²) = √(9 + 16 + 144) = √169.
Answer: 13
Example 2: Find the unit vector in the direction of v = (6, −8).
|v| = √(36 + 64) = √100 = 10.
v̂ = (6/10, −8/10).
Answer: (3/5, −4/5)
Example 3: Find the distance between the points A(1, 2, 3) and B(4, 6, 8).
Vector AB = (4−1, 6−2, 8−3) = (3, 4, 5).
Distance = |AB| = √(9 + 16 + 25) = √50.
Answer: 5√2
Sample MCQs
1. A scalar quantity has:
a) Magnitude and direction b) Only magnitude c) Only direction d) Neither magnitude nor direction
Answer: b) Only magnitude
2. A vector quantity has:
a) Only magnitude b) Only direction c) Both magnitude and direction d) Neither
Answer: c) Both magnitude and direction
3. The magnitude of the vector u = (3, 4) is:
a) 3 b) 4 c) 5 d) 7
Answer: c) 5
4. A unit vector has magnitude:
a) 0 b) 1 c) Equal to the original vector d) Undefined
Answer: b) 1
5. Two vectors are parallel if:
a) They have the same magnitude b) They are non-zero scalar multiples of each other c) They are perpendicular d) They are equal
Answer: b) They are non-zero scalar multiples of each other
Important Short Questions
- Define a scalar quantity and give two examples.
- Define a vector quantity and give two examples.
- Define the magnitude of a vector u = (x, y, z).
- Define a unit vector, and state the formula for finding one from a given vector.
- Define a position vector.
Important Long Questions
- Find the magnitude of the vector u = (2, −3, 6), and find the unit vector in its direction.
- Find the distance between the points A(2, −1, 4) and B(5, 3, −2).
- Explain the difference between equal vectors and parallel vectors, giving an example of each.
- If u = (1, 2, 3) and v = (2, 4, 6), determine whether u and v are parallel, explaining your reasoning.
How to Approach This Exercise Effectively
- Always compute the magnitude first when a problem asks for a unit vector — every unit vector calculation depends on it.
- Remember equal vectors need both matching magnitude AND matching direction — matching magnitude alone isn’t enough.
- To check if two vectors are parallel, see if one is a constant multiple of the other component-by-component.
- Treat ‘distance between two points’ and ‘magnitude of the vector between them’ as the same calculation — they always give the same answer.
- Keep scalar and vector quantities clearly labeled in your own notes and examples — mixing them up is a common early mistake.
FAQs
Q: What is the difference between a scalar and a vector?
A: A scalar has only a size (magnitude), while a vector has both a size and a specific direction.
Q: Why is a unit vector useful?
A: It captures pure direction information, with the ‘size’ factored out, which is useful whenever only the direction of a quantity matters.
Q: Is the zero vector considered parallel to every other vector?
A: By convention, the zero vector is often treated as parallel to every vector, since parallelism is usually only meaningfully discussed for non-zero vectors.
Q: How is distance between two points related to vector magnitude?
A: The distance between two points equals the magnitude of the vector that goes from one point to the other.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 14: Vectors in Space, Exercise 14.1.