- The length of the cross product of two vectors
- The scalar triple product of the vectors a, b, and c
- The volume of the parallelepiped determined by the vectors a, b, and c is the magnitude of their scalar triple product.
Explanation:
- The length of the cross product of two vectors is
| a [tex]\times[/tex] b | = |a| |b| sin θ
- The length of the cross product of two vectors is equal to the area of the parallelogram determined by the two vectors (see figure below).
| a [tex]\times[/tex] b | = - | b [tex]\times[/tex] a |
- Multiplication by scalars:
(ca) [tex]\times[/tex] b = c (a [tex]\times[/tex] b) = a [tex]\times[/tex] (cb)
a [tex]\times[/tex] (b + c) = (a [tex]\times[/tex] b) + (a [tex]\times[/tex] c)
- The scalar triple product of the vectors a, b, and c:
a . (b [tex]\times[/tex] c) = (a [tex]\times[/tex] b) . c
- The magnitude of the scalar triple product is the volume of the parallelepiped of the vectors a, b, and c.
- The vector triple product of the vectors a, b, and c is given as
a [tex]\times[/tex] (b [tex]\times[/tex] c) = (a.c) b - (a.b) [tex]\times[/tex]c
