Cross Product

Definition - 3-space

Only defined in 3-space, produces a vector perpendicular to and

Given and , then

Definition

Intuition

Recall that the determinant is the area of a parallelogram. That means the cross-product is also the area of a parallelogram (see below).

Theorem

For any 2 non-zero vectors in 3-space, if , then and are colinear

Properties

Application - Area of a Parallelogram

Area of a Parallelogram.png
We see that and satisfies , which gives

And the area of a triangle is calculated as half the area of a paralellogram

Volume of a Parallelepiped:
Dot product the cross product VECTOR

Hand Rule

Figure

The resulting direction of :

The left hand rule is more "intuitive" as the first vector is the bottom finger, while the right hand needs to mimic the left hand by swapping fingers.

Lagrange Identity

Identity

Where

Cross Product Order

Definition

In Cartesian coordinates:

In Cylindrical Coordinates:

In Spherical Coordinates:

What this means:

The same applies to the rest of the coordinate systems.