Let the vectors a , b , c represent three coterminous edges of a parallelepiped of volume V. Then the volume of the parallelepiped whose coterminous edges are represented by a , b+ c and a + 2 b + 3 c is equal to [2023]

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Solution

Q.
Let the vectors $\vec{a}, \vec{b}, \vec{c}$ represent three coterminous edges of a parallelepiped of volume V. Then the volume of the parallelepiped whose coterminous edges are represented by $\vec{a}, \vec{b} + \vec{c}$ and $\vec{a} + 2 \vec{b} + 3 \vec{c}$ is equal to [2023]

1 $V$

2 $3 V$

3 $2 V$

4 $6 V$

Ans.
(1)

We have,

$Volume of parallelepiped = [\vec{a} \vec{b} \vec{c}] = V$ ...(i)

New volume of parallelepiped $= [\vec{a} \vec{b} + \vec{c} \vec{a} + 2 \vec{b} + 3 \vec{c}]$

$=$ $| \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 2 & 3 \end{matrix} |$ $[\vec{a} \vec{b} \vec{c}] = {1 (3 - 2) - 0 + 0} [\vec{a} \vec{b} \vec{c}]$

$= [\vec{a} \vec{b} \vec{c}] = V$ $(Using (i))$

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