Cross (vector) Product

Cross (vector) Product

What is it?

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.

In simple words

• 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.

Important terms

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 = (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.

What you should learn

• 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.

Examples

• Right hand
• a x a = 0
• Flip order, flip sign

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|.

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.

Full school-level explanation

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.

Size and direction, then i j k:

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

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|.

Remember

• area = length of cross

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.

Remember

• cyclic plus

Examples

• i x j = k
• j x i = -k
• i x i = 0

How to answer a cross (vector) product question

How it works

• 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.

Remember

• |a x b| = |a||b| sin theta
• direction: right hand
• 2
• a x b = - b x a
• area = length of cross
• cyclic plus
• |a x b| = |a||b| sin theta
• direction: right hand

How the ideas connect

Worked: |i x (2j)|. i x 2j = 2 (i x j) = 2k |2k| = 2 also |i||2j| sin 90 = 1*2*1 = 2

Deeper understanding

Go beyond memorising Cross (vector) Product. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: |i x (2j)|. i x 2j = 2 (i x j) = 2k |2k| = 2 also |i||2j| sin 90 = 1*2*1 = 2

Common mistakes

Exam focus

Core Idea

1. GEOMETRY

2. i j k TABLE

Worked Example

Exam focus

Quick revision

• |a x b| = |a||b| sin theta
• direction: right hand
• 2
• a x b = - b x a
• CROSS (VECTOR) PRODUCT: a x b is a vector, perpendicular to both a and b, length |a||b| sin theta (area of the parallelogram
• 1. GEOMETRY: Area of parallelogram = |a x b|. Triangle area = (12)|a x b|. Parallel vectors: a x b = 0. Unit nor

Cross is not commutative. Swapping order changes sign.

For long answers on Cross (vector) Product: define clearly → explain the main parts → give one example → state one exam caution.