System of particles and Rotational motion NCERT Solutions Class 11 Physics - Solved Exercise Question 7.5

Question 7.5:

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

Solution:
A parallelepiped with origin O and sides a, b, and c is shown in the following figure.

Volume of the given parallelepiped = abc
OC = a
OB = b
OC = c
Let n^ be a unit vector perpendicular to both b and c. Hence, n^ and a have the same direction.
∴ b X c = bc Sin θ n^
= bc n^
a.(b X c) = a(bcn^)
= abc Cosθ n^
= abc Cos 0
= abc

= Volume of the parallelepiped