Question

Cross product and its properties

Answer 1

• 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 $\times$ 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).

• Anticommutativity:

                                      | a $\times$ b | = - | b $\times$ a |

• Multiplication by scalars:

                                (ca) $\times$ b = c (a $\times$ b) = a $\times$ (cb)

• Distributivity:  

                                  a $\times$ (b + c) = (a $\times$ b) + (a $\times$ c)

• The scalar triple product of the vectors a, b, and c:

                                      a . (b $\times$ c) = (a $\times$ 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 $\times$ (b $\times$ c) = (a.c) b - (a.b) $\times$c