BeginnerTopic 20 of 104Engineering Mathematics
Cross product
a×b gives a vector perpendicular to both, with magnitude equal to the parallelogram area; used for torque.
In depth
The cross product a×b is a vector perpendicular to both with magnitude ‖a‖‖b‖sinθ and direction by the right-hand rule. It is anti-commutative. Torque τ=r×F, angular momentum L=r×p and the Lorentz force use it.
Key points
- Determinant form for computing
- Zero if vectors are parallel
- Magnitude equals parallelogram area
ExampleA 50 N force 0.4 m from a pivot at 90° gives torque 20 N·m.
What is Cross product?
a×b gives a vector perpendicular to both, with magnitude equal to the parallelogram area; used for torque.
Why is it important to understand cross product?
The cross product gives a vector normal to a plane and a torque or magnetic force direction, which is essential in rotating and electromagnetic systems.
Why is the cross product not commutative?
Reversing the order reverses the direction, so a×b equals minus b×a. Keep track of the sign when you compute torque or force.
How is the cross product's magnitude related to area?
It equals the area of the parallelogram spanned by the two vectors, which gives the area of a triangle as half that value.
How is Cross product used in practice?
Typical applications include computing torque from a force applied at a lever arm, finding the magnetic force on a current-carrying wire and finding the normal vector of a triangular face in a 3D model.
What should I learn before Cross product?
Study "Dot product" first (topic 19 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Cross product?
Determinant form for computing. Zero if vectors are parallel. Magnitude equals parallelogram area.
Can you give an example of Cross product?
A 50 N force 0.4 m from a pivot at 90° gives torque 20 N·m.
What comes after Cross product?
The next topic is "Sequences and series", which builds directly on this one.
How difficult is Cross product?
It is classed as Beginner in Engineering Mathematics (topic 20 of 104). It is an entry-level topic. It relies only on basic school mathematics and the ideas introduced earlier in this discipline.