Geometric Applications to the Cross Product

There are a so many geometric applications to the cross product also.  Assume we have three vectors a^→, b^→ and c^→ and we make the three dimensional figure displayed below.

The area of the above parallelogram that is two dimensional front of this object is defined by,

Area = ||a^→ * b^→||

and the volume of the parallelepiped that is the whole three dimensional object is defined by,

Volume = |a^→ • (b^→ * c^→)|

Notice: The absolute value bars are needed as the quantity could be negative and volume isn't negative.

 We can make use of this volume fact to determine if three vectors lie in similar plane or not.  If three vectors lie in similar plane then the volume of the parallelepiped will be zero.