Question

Show that a. (b × c) is equal in magnitude to the volume of the parallelepiped formed on the three vectors, a, b and c.

Answer 1

See the figure ,
A parallelepiped be formed of three vectors ,
OA = a
OB = b
OC = c

Cross product of vector b and c is (b×c) = bcsin90° = bcñ
Here ñ is a unit vector along OA , e.g unit vector along perpendicular to the plane containing b and c
Now,
a.(b×c) = a.(bcñ)
= a(bc)cos0°
= abc = Length × width × height
= volume of parallelepiped

Hence, proved//