Next: The box product. Up: Analytic geometry in the Previous: Dot product.

# Cross product.

Definition 1.17   Let and be two vectors in the space.
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;
• .

Definition 1.18   A triple of vectors which satisfies the right-hand rule is called direct.

Proposition 1.19

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

Remark 1.20

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

Then:

Proposition 1.21 (Determinant Formula)

Example 1.22        Take and . Then:

Example 1.23   Find a unit normal vector for the plane through the points A(1,2,-1), B(2,0,1) and C(-1,2,4).

The vector is noemal to ; we have:

The norm of this vector is equal to:

The plane has two unit normal vectors, namely:

Next: The box product. Up: Analytic geometry in the Previous: Dot product.
Noah Dana-Picard
2001-05-30