1st Year Math Chapter 14 Exercise 14.3 Notes: The Cross 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 14.3 introduces the cross product (or vector product) of two vectors in space, which — unlike the dot product — produces another vector, perpendicular to both original vectors.
What Does This Exercise Cover?
This exercise teaches how to compute the cross product using the determinant formula, how to test whether two vectors are parallel, and how to use the cross product to find areas and the moment of a force.
Key Concepts
1. Definition of the Cross Product
For two vectors u and v in space, the cross product is u × v = |u||v| sin θ · n̂, where θ is the angle between u and v (0 ≤ θ ≤ π), and n̂ is a unit vector perpendicular to both, following the right-hand rule. The cross product is anti-commutative: u × v = −(v × u).
2. Cross Products of the Unit Vectors i, j, k
i × i = 0, j × j = 0, k × k = 0 (a vector crossed with itself gives the zero vector), while i × j = k, j × k = i, k × i = j.
3. The Determinant Formula
If u = a₁i + b₁j + c₁k and v = a₂i + b₂j + c₂k, then u × v is computed as the determinant of a 3×3 matrix with i, j, k in the first row and the components of u and v in the next two rows.
4. Parallel Vectors and Angle Between Two Vectors
Two vectors are parallel if and only if u × v = 0 (the zero vector). The angle between two vectors can also be found using sin θ = |u × v| / (|u||v|).
5. Areas and Moment of Force
The area of a parallelogram with adjacent sides u and v is |u × v|; the area of the corresponding triangle is half that: (1/2)|u × v|. The moment (torque) of a force F acting at position vector r about the origin is r × F.
Solved Examples
Example 1: Find u × v where u = (1, 0, 0) and v = (0, 1, 0).
Using the determinant formula: u × v = i(0×0 − 0×1) − j(1×0 − 0×0) + k(1×1 − 0×0).
= i(0) − j(0) + k(1).
Answer: (0, 0, 1)
Example 2: Find the area of the parallelogram formed by u = (2, 0, 0) and v = (0, 3, 0).
u × v = i(0×0 − 0×3) − j(2×0 − 0×0) + k(2×3 − 0×0) = (0, 0, 6).
Area = |u × v| = |(0,0,6)|.
Answer: 6 square units
Example 3: Determine whether u = (2, 4, 6) and v = (1, 2, 3) are parallel.
Notice u = 2v, since 2(1,2,3) = (2,4,6).
Since u is a scalar multiple of v, they are parallel, and their cross product should equal the zero vector.
Answer: u and v are parallel
Sample MCQs
1. The cross product of two vectors u and v is:
a) A scalar b) A vector perpendicular to both u and v c) Always zero d) Undefined
Answer: b) A vector perpendicular to both u and v
2. The cross product i × j equals:
a) 0 b) k c) −k d) i
Answer: b) k
3. Two vectors are parallel if their cross product equals:
a) 1 b) The zero vector c) Their magnitudes d) −1
Answer: b) The zero vector
4. The area of a parallelogram with adjacent sides u and v is:
a) u·v b) |u × v| c) |u| + |v| d) (1/2)|u × v|
Answer: b) |u × v|
5. The cross product is:
a) Commutative b) Anti-commutative c) Always positive d) Always a scalar
Answer: b) Anti-commutative
Important Short Questions
- State the definition of the cross product of two vectors.
- State the cross products i × j, j × k, and k × i.
- State the condition for two vectors to be parallel, in terms of the cross product.
- Write the formula for the area of a triangle using the cross product of two of its sides.
- Explain why the cross product of two vectors is anti-commutative.
Important Long Questions
- Find u × v where u = (1, 2, 3) and v = (4, 5, 6), using the determinant formula.
- Find the area of the parallelogram formed by the vectors u = (3, 0, 0) and v = (0, 4, 0).
- Determine whether u = (2, 4, 6) and v = (1, 2, 3) are parallel, using the cross product.
- Explain the geometric meaning of |u × v|, connecting it to the area of a parallelogram.
How to Approach This Exercise Effectively
- Always set up the determinant with i, j, k in the top row and the two vectors’ components below — a consistent layout avoids sign mistakes.
- Double-check your signs when expanding the determinant — the j-component always carries a negative sign in the standard expansion.
- Remember the cross product result is itself a vector — don’t accidentally report just its magnitude when a full vector answer is expected.
- For area problems, compute the cross product first, then take its magnitude — the order matters, since |u|×|v| is not the same as |u × v|.
- Practice recognizing parallel vectors quickly by inspection (checking if one is a scalar multiple of the other) before resorting to the full cross product calculation.
FAQs
Q: Why is the cross product a vector, unlike the dot product?
A: By definition, the cross product produces a new vector perpendicular to both original vectors, capturing both a magnitude (related to the area they span) and a direction, whereas the dot product only produces a single number.
Q: What does the ‘right-hand rule’ refer to?
A: It’s a convention for determining the direction of the cross product’s result: curling the fingers of the right hand from the first vector toward the second, the thumb points in the direction of u × v.
Q: How is the cross product used to find the area of a triangle?
A: The area of a triangle formed by two vectors along its sides is exactly half the area of the parallelogram they would form, so Area = (1/2)|u × v|.
Q: What is a real-world example that uses the cross product?
A: The moment (or torque) of a force about a point is calculated as r × F, where r is the position vector to the point of application and F is the force.
Notes prepared for Punjab Board (PECTAA) 1st Year Mathematics, 2026-27 session, Chapter 14: Vectors in Space, Exercise 14.3.