1st Year Math Chapter 14 Exercise 14.2 Notes: The Dot Product (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.
Exercise 14.2 introduces the dot product (or scalar product) of two vectors, a way of combining two vectors to produce a single number that reveals the angle between them.
What Does This Exercise Cover?
This exercise teaches the geometric and component definitions of the dot product, how to test whether two vectors are perpendicular, how to find the angle between two vectors, and a real-world application: work done by a constant force.
Key Concepts
1. Definition of the Dot Product
For two non-zero vectors u and v, the dot product is defined as u·v = |u||v| cos θ, where θ is the angle between them (0 ≤ θ ≤ π).
2. Dot Products of the Unit Vectors i, j, k
i·i = 1, j·j = 1, k·k = 1 (each unit vector dotted with itself gives 1), while i·j = 0, j·k = 0, k·i = 0 (different unit vectors are perpendicular, so their dot product is 0).
3. Component Formula
If u = (a₁, b₁, c₁) and v = (a₂, b₂, c₂), then u·v = a₁a₂ + b₁b₂ + c₁c₂. The dot product is commutative: u·v = v·u.
4. Perpendicular Vectors and the Angle Between Two Vectors
Two non-zero vectors are perpendicular (orthogonal) if and only if u·v = 0. More generally, the angle between two vectors is found using cos θ = (u·v) / (|u||v|).
5. Work Done by a Constant Force
If a constant force F moves a body through a displacement d, the work done is W = F·d = |F||d| cos θ, where θ is the angle between the force and the direction of motion.
Solved Examples
Example 1: Find the dot product of u = (2, 3, −1) and v = (4, −1, 5).
u·v = 2(4) + 3(−1) + (−1)(5) = 8 − 3 − 5.
Answer: 0 (so u and v are perpendicular)
Example 2: Find the angle between u = (1, 0, 1) and v = (0, 1, 1).
u·v = (1)(0) + (0)(1) + (1)(1) = 1.
|u| = √2, |v| = √2.
cos θ = 1 / (√2 × √2) = 1/2.
Answer: θ = 60°
Example 3: A constant force F = (3, 4) newtons moves an object through a displacement d = (5, 0) meters. Find the work done.
W = F·d = 3(5) + 4(0).
Answer: 15 joules
Sample MCQs
1. The dot product of two vectors is defined as:
a) |u||v| sin θ b) |u||v| cos θ c) |u| + |v| d) |u| − |v|
Answer: b) |u||v| cos θ
2. The dot product i·j equals:
a) 1 b) 0 c) −1 d) Undefined
Answer: b) 0
3. Two vectors are perpendicular if their dot product equals:
a) 1 b) 0 c) −1 d) Their magnitudes
Answer: b) 0
4. For u = (1, 2, 3) and v = (4, 5, 6), u·v equals:
a) 30 b) 32 c) 12 d) 21
Answer: b) 32
5. Work done by a constant force is calculated using:
a) The cross product b) The dot product c) The magnitude only d) The unit vector
Answer: b) The dot product
Important Short Questions
- State the definition of the dot product of two vectors.
- State the dot products of the unit vectors i, j, and k with themselves and with each other.
- Find the dot product of u = (2, −1, 3) and v = (1, 4, −2).
- State the condition for two vectors to be perpendicular, in terms of the dot product.
- Write the formula for finding the angle between two vectors using the dot product.
Important Long Questions
- Find the dot product of u = (3, −2, 1) and v = (2, 1, −4), and determine whether they are perpendicular.
- Find the angle between the vectors u = (1, 0, 1) and v = (0, 1, 1).
- A constant force F = (2, 5) newtons moves an object through a displacement d = (4, 3) meters. Find the work done by the force.
- Explain why the dot product of two vectors is a scalar, not a vector.
How to Approach This Exercise Effectively
- Use the component formula (a₁a₂ + b₁b₂ + c₁c₂) as your default calculation method — it’s faster than the angle-based definition for most problems.
- Whenever a dot product comes out to 0, immediately note the vectors are perpendicular — this is a frequent shortcut in longer problems.
- For angle-between-vectors questions, compute u·v, |u|, and |v| separately before combining them in the cosine formula.
- Remember the dot product formula for work only applies to a constant force over a straight-line displacement.
- Keep dot product results (scalars) visually distinct from vector results in your working, to avoid confusing this exercise with cross products in Exercise 14.3.
FAQs
Q: Why is the dot product also called the ‘scalar product’?
A: Because the result of a dot product is always a single number (a scalar), not another vector.
Q: What does it mean if the dot product of two vectors is negative?
A: It means the angle between the vectors is obtuse (greater than 90°), since cosine is negative for angles between 90° and 180°.
Q: How is the dot product used to find the angle between two vectors?
A: Rearranging u·v = |u||v|cos θ gives cos θ = (u·v)/(|u||v|), which can then be used to find θ directly.
Q: What real-world quantity is calculated using the dot product of force and displacement?
A: Work done by a force is calculated exactly this way — as the dot product of the force vector and the displacement vector.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 14: Vectors in Space, Exercise 14.2.