MathematicsChapter 3: Vectors

Cross Product in Components

a x b = det of a 3x3 with row 1 = i j k, row 2 = components of a, row 3 = components of b (school mnemonic).

Learning objectives

State the main ideas and notation used in cross product in components.
Apply the rules of cross product in components in a clear, step-by-step solution.
Check signs, conditions, and the final answer for mathematical accuracy.

Definition

a x b = det of a 3x3 with row 1 = i j k, row 2 = components of a, row 3 = components of b (school mnemonic). Expand: (a2 b3 - a3 b2) i - (a1 b3 - a3 b1) j + (a1 b2 - a2 b1) k. Distributive: a x (b+c) = a x b + a x c. Angle: sin theta = |a x b| / (|a||b|). Meaning: the determinant is just a compact way to write those three 2x2 minors.

Mathematical form

b = det of a 3x3 with row 1

theta =

Where / symbols

CROSS PRODUCT IN COMPONENTS: a x b = det of a 3x3 with row 1 = i j k, row 2 = components of a, row 3 = components of b (school mnemonic). Expand: (a2 b3 - a3 b2) i - (a1 b3 - a3 b1) j + (a1 b2 - a2 b1) k. Distributive: a x (b+c) = a x b + a x c. Angle: sin theta = |a x b| / (|a||b|). Meaning: the determinant is just a compact way to write those three 2x2 minors.
1. THE DETERMINANT: Write i j k on top. Compute the three cofactors. Watch the minus in front of j. Practice one fully written expansion.
2. USES: Area. Torque tau = r x F named (physics). Perp vector to a plane of two vectors. If the result is 0, the vectors are parallel (or one is 0).

Overview

Expand, then the angle:

The Determinant

Write i j k on top. Compute the three cofactors. Watch the minus in front of j. Practice one fully written expansion.

Examples i (a2b3-a3b2) · - j (a1b3-a3b1) · k (a1b2-a2b1)

Uses

Area. Torque tau = r x F named (physics). Perp vector to a plane of two vectors. If the result is 0, the vectors are parallel (or one is 0).

Examples sin theta from |cross| · Check a · (a x b) = 0 · Distributive

WHAT — CROSS PRODUCT IN COMPONENTS

RULE — 1. THE DETERMINANT

Write i j k on top. Compute the three cofactors. Watch the minus in front of j. Practice one fully written expansion.

METHOD — 2. USES

Area. Torque tau = r x F named (physics). Perp vector to a plane of two vectors. If the result is 0, the vectors are parallel (or one is 0).

WORKED EXAMPLE

Worked: <1,2,3> x <4,5,6>. i (12-15) - j (6-12) + k (5-8) i (-3) - j (-6) + k (-3) = -3 i + 6 j - 3 k

How to answer a cross product in components 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

<1,2,3> x <4,5,6>.
i (12-15) - j (6-12) + k (5-8)
i (-3) - j (-6) + k (-3)
= -3 i + 6 j - 3 k

In simple words

The j-component formula already includes the minus from the cofactor. Do not minus twice.

Key points

ijk determinant mnemonic
middle component has a minus
minus on j
area and normal

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.