- 1.
- If at least one of them is equal to or if they are linearly dependent (parallel), then we define: .
- 2.
- Suppose that none of the given vectors is equal to
and that these vectors are linearly
independent. Denote by
the (oriented) angle from
to
.
The two given vectors determine a plane
;
denote by
a vector, orthogonal to the
plane
and verifying the
*right-hand*rule, i.e. if the right-hand thumb points in the direction of , then the fingers curl through the angle from to . The cross product is the vector defined by the two following properties:- is parallel to and points towards the same direction;
- .

- 1.
- , .
- 2.
- ,
- 3.
- , .
- 4.
- , .
- 5.
- , ,

Choose now a direct (v.s. Def. 1.18) orthonormal basis
for the 3-dimensional space. Denote:

Then:

The vector
is noemal to
;
we have:

The norm of this vector is equal to:

The plane has two unit normal vectors, namely: