MathematicsChapter 3: Vectors
Cross (vector) Product
a x b is a vector, perpendicular to both a and b, length |a||b| sin theta (area of the parallelogram they span).
Learning objectives
- State the main ideas and notation used in cross (vector) product.
- Apply the rules of cross (vector) product in a clear, step-by-step solution.
- Check signs, conditions, and the final answer for mathematical accuracy.
Definition
a x b is a vector, perpendicular to both a and b, length |a||b| sin theta (area of the parallelogram they span). Direction: right-hand rule. a x a = 0. a x b = - b x a (anticommutative). i x j = k, j x k = i, k x i = j, and i x i = 0. Meaning: the cross builds a new arrow normal to a plane.
Mathematical form
a = 0. a x b
j = k
k = i
i = j
Where / symbols
- CROSS (VECTOR) PRODUCT
- a x b is a vector, perpendicular to both a and b, length |a||b| sin theta (area of the parallelogram they span). Direction: right-hand rule. a x a = 0. a x b = - b x a (anticommutative). i x j = k, j x k = i, k x i = j, and i x i = 0. Meaning: the cross builds a new arrow normal to a plane.
- 1. GEOMETRY
- Area of parallelogram = |a x b|. Triangle area = (1/2)|a x b|. Parallel vectors: a x b = 0. Unit normal n-hat = (a x b)/|a x b|.
- 2. i j k TABLE
- Cyclic i->j->k->i. Opposite cycle gives a minus. These three lines are the exam memory: i x j = k, j x k = i, k x i = j.
Relationship
- |a x b| = |a||b| sin theta
Overview
Size and direction, then i j k:
Geometry
Area of parallelogram = |a x b|. Triangle area = (12)|a x b|. Parallel vectors: a x b = 0. Unit normal n-hat = (a x b)/|a x b|.
Examples Right hand · a x a = 0 · Flip order, flip sign
i j k TABLE
Cyclic i->j->k->i. Opposite cycle gives a minus. These three lines are the exam memory: i x j = k, j x k = i, k x i = j.
Examples i x j = k · j x i = -k · i x i = 0
WHAT — CROSS (VECTOR) PRODUCT
a x b is a vector, perpendicular to both a and b, length |a||b| sin theta (area of the parallelogram they span). Direction: right-hand rule. a x a = 0. a x b = - b x a (anticommutative). i x j = k, j x k = i, k x i = j, and i x i = 0. Meaning: the cross builds a new arrow normal to a plane.
RULE — 1. GEOMETRY
Area of parallelogram = |a x b|. Triangle area = (12)|a x b|. Parallel vectors: a x b = 0. Unit normal n-hat = (a x b)/|a x b|.
METHOD — 2. i j k TABLE
Cyclic i->j->k->i. Opposite cycle gives a minus. These three lines are the exam memory: i x j = k, j x k = i, k x i = j.
WORKED EXAMPLE
Worked: |i x (2j)|. i x 2j = 2 (i x j) = 2k |2k| = 2 also |i||2j| sin 90 = 1*2*1 = 2
How to answer a cross (vector) product question
- Identify the given quantities, notation, and required result.
- Choose the matching rule and write it before substitution.
- Work one algebraic step at a time, keeping brackets and signs clear.
- Check restrictions, simplify the result, and verify it where possible.
Worked example
- |i x (2j)|.
- i x 2j = 2 (i x j) = 2k
- |2k| = 2
- also |i||2j| sin 90 = 1*2*1 = 2
In simple words
Cross is not commutative. Swapping order changes sign.
Key points
- |a x b| = |a||b| sin theta
- direction: right hand
- area = length of cross
- cyclic plus
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Cross (vector) Product
Sticky Notes · Quick Revision
Core Idea
- a x b is a vector, perpendicular to both a and b, length |a||b| sin theta (area of the parallelogram they span).
- Direction: right-hand rule.
- a x a = 0.
1. GEOMETRY
- Area of parallelogram = |a x b|.
- Triangle area = (12)|a x b|.
- Parallel vectors: a x b = 0.
- Unit normal n-hat = (a x b)/|a x b|.
2. i j k TABLE
- Cyclic i->j->k->i.
- Opposite cycle gives a minus.
- These three lines are the exam memory: i x j = k, j x k = i, k x i = j.
Worked Example
- Worked: |i x (2j)|.
- i x 2j = 2 (i x j) = 2k
- |2k| = 2
- also |i||2j| sin 90 = 1*2*1 = 2
Common Misconception
- Cross is not commutative. Swapping order changes sign.
FBISE Exam Check
- State the rule for cross (vector) product before substituting values.
- Show the main algebra steps, keep exact values where possible, and check every condition.
60-second revision
- a x b is a vector, perpendicular to both a and b, length |a||b| sin theta (area of the parallelogram they span).
- Direction: right-hand rule.
- a x a = 0.
- Cross is not commutative. Swapping order changes sign.
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