Question

If the vectors 2î - 3ĵ, î + ĵ - k̂ and 3î - k̂ form the three co-terminous edges of a parallelepiped, then the volume of parallelepiped is:
1. 8
2. 10
3. 4
4. 14

Answer 1

Correct Answer - Option 3 : 4

Concept:

For two vectors \(\rm \vec A\) and \(\rm \vec B\) at an angle θ to each other:

Dot Product is defined as \(\rm \vec A.\vec B=|\vec A||\vec B|\cos \theta\).

Cross Product is defined as \(\rm \vec A\times \vec B=\vec n|\vec A||\vec B|\sin \theta\) where \(\rm \vec n\) is the unit vector perpendicular to the plane containing \(\rm \vec A\) and \(\rm \vec B\).

 

For three vectors \(\rm \vec A\), \(\rm \vec B\) and \(\rm \vec C\):

Triple Cross Product: is defined as:

 \(\rm \vec A\times(\vec B\times\vec C)=(\vec A.\vec C)\vec B-(\vec A.\vec B)\vec C\).

Triple Scalar Product (Box Product): is defined as 

\(\rm [\vec A\ \vec B\ \vec C]=\vec A.(\vec B\times\vec C)=\begin{vmatrix} \rm a_1 & \rm a_2 & \rm a_3 \\ \rm b_1 & \rm b_2 & \rm b_3 \\\rm c_1 & \rm c_2 & \rm c_3 \end{vmatrix}\).

Volume of a parallelepiped, with vectors \(\rm \vec a\), \(\rm \vec b\) and \(\rm \vec c\) as its sides, is given by the box product of the three vectors.

Volume = \(\rm [\vec a\ \vec b\ \vec c]\).

Calculation:

Let's say that the sides of the parallelepiped are:

\(\rm \vec a\) = 2î - 3ĵ + 0k̂

\(\rm \vec b\) = î + ĵ - k̂

\(\rm \vec c\) = 3î + 0ĵ - k̂

∴ Volume = \(\rm \rm [\vec a\ \vec b\ \vec c]=\begin{vmatrix} 2 & -3 & \ \ \ 0 \\ 1 & \ \ \ 1 & -1 \\ 3 & \ \ \ 0 & -1 \end{vmatrix}\) 

= 2(-1 - 0) - 3(-3 + 1) + 0(0 - 3)

= -2 + 6 + 0 = 4.