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Advanced ▦ Linear Algebra Topic 11 of 104

Cross Product

A vector perpendicular to two vectors in 3D.

Key idea

A vector perpendicular to two vectors in 3D.

In depth

For 3D vectors a × b is perpendicular to both, with length |a||b| sin θ equal to the parallelogram area. Direction follows the right-hand rule, and a × b = −(b × a).

Worked example

(1, 2, 3) × (4, 5, 6) = (2·6 − 3·5, 3·4 − 1·6, 1·5 − 2·4) = (−3, 6, −3).

Formula

u×v

Real-world uses

Common mistake

Expecting commutativity, or using the cross product in 2D without lifting to 3D.

How to learn this

  1. Read the key idea and say it in your own words.
  2. Work through the worked example, then try a similar problem.
  3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

What is Cross Product?

A vector perpendicular to two vectors in 3D. It is a topic at the advanced level in Linear Algebra.

What is the formula for Cross Product?

The key formula is u×v.

Where is Cross Product used in real life?

Real-world examples: Finding the axis of rotation for a spinning wheel; The normal direction of a surface for 3D graphics.

What should I learn before Cross Product?

Make sure you are comfortable with Projection, the previous topic in Linear Algebra.

What is a common mistake with Cross Product?

Expecting commutativity, or using the cross product in 2D without lifting to 3D.

How can I practise Cross Product?

Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.

What comes after Cross Product?

Move on to Triple Product next.

Related advanced topics in Linear Algebra