Cross Product (3D)
A vector perpendicular to two vectors.
Key idea
A vector perpendicular to two vectors.
In depth
a × b is a vector perpendicular to both, with magnitude |a||b| sin θ equal to the area of the parallelogram they span. Direction follows the right-hand rule, and it flips if the order swaps.
Worked example
(1,0,0) × (0,1,0) = (0,0,1).
Formula
Real-world uses
- Finding the normal to a surface for lighting in 3D graphics
- Torque in engineering
Common mistake
Assuming a × b = b × a. It equals −(b × a).
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Cross Product (3D)?
A vector perpendicular to two vectors. It is a topic at the expert level in Geometry.
What is the formula for Cross Product (3D)?
The key formula is |a×b| = |a||b|sinθ.
Where is Cross Product (3D) used in real life?
Real-world examples: Finding the normal to a surface for lighting in 3D graphics; Torque in engineering.
What should I learn before Cross Product (3D)?
Make sure you are comfortable with Dot Product (Geometric), the previous topic in Geometry.
What is a common mistake with Cross Product (3D)?
Assuming a × b = b × a. It equals −(b × a).
How can I practise Cross Product (3D)?
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 (3D)?
Move on to 3D Coordinates next.
Related expert topics in Geometry
- Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
- Cyclic Quadrilaterals – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
- Cavalieri’s Principle – Solids with equal cross-sectional areas at every height have equal volume.
- Area by Coordinates (Shoelace) – Area of a polygon from vertices.
- Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
- Ellipse – Sum of distances to two foci is constant.