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Question: If the vectors \(2 \mathbf { i } - 3 \mathbf { j } , \mathbf { i } + \mathbf { j } - \mathbf { k }\...

If the vectors 2i−3j,i+j−k2 \mathbf { i } - 3 \mathbf { j } , \mathbf { i } + \mathbf { j } - \mathbf { k } and 3i−k3 \mathbf { i } - \mathbf { k } form three concurrent edges of a parallelopiped, then the volume of the parallelopiped is

A

8

B

10

C

4

D

14

Answer

4

Explanation

Solution

Here, OA→=2i−3j=a\overrightarrow { O A } = 2 \mathbf { i } - 3 \mathbf { j } = \mathbf { a } (say)

OB→=i+j−k=b\overrightarrow { O B } = \mathbf { i } + \mathbf { j } - \mathbf { k } = \mathbf { b }(say)

and OC→=3i−k=c\overrightarrow { O C } = 3 \mathbf { i } - \mathbf { k } = \mathbf { c } (say)

Hence volume is [abc]=a⋅(b×c)=∣2−3011−130−1∣=4[ \mathbf { a } \mathbf { b } \mathbf { c } ] = \mathbf { a } \cdot ( \mathbf { b } \times \mathbf { c } ) = \left| \begin{array} { c c c } 2 & - 3 & 0 \\ 1 & 1 & - 1 \\ 3 & 0 & - 1 \end{array} \right| = 4 .