Let the vectors a → ,   b → ,   c → represent three coterminous edges of a parallelopiped of volume V . Then the volume of the parallelopiped, whose coterminous edges are represented by a → ,   b → + c → and a → + 2 b → + 3 c →…

Question

Let the vectors a &#8594; , &#160; b &#8594; , &#160; c &#8594; represent three coterminous edges of a parallelopiped of volume V . Then the volume of the parallelopiped, whose coterminous edges are represented by a &#8594; , &#160; b &#8594; + c &#8594; and a &#8594; + 2 b &#8594; + 3 c &#8594; is equal to

Aicharya · Accepted Answer

We know that,
Volume of parallelepiped whose edges determined by vectors $\vec{a}, \vec{b}, \vec{c}$ is $[\begin{matrix} \vec{a} & \vec{b} & \vec{c} \end{matrix}]$ .
Now it is given that $V$ is volume of parallelepiped $\vec{a}, \vec{b}, \vec{c}$ , so required volume of parallelepiped with new edges will be
$= [\begin{matrix} \vec{a} & \vec{b} + \vec{c} & \vec{a} + 2 \vec{b} + 3 \vec{c} \end{matrix}]$
$= |\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 2 & 3 \end{matrix}| [\begin{matrix} \vec{a} & \vec{b} & \vec{c} \end{matrix}]$
$= \{(1) (3 - 2) - 0 + 0\} [\begin{matrix} \vec{a} & \vec{b} & \vec{c} \end{matrix}]$
$= [\begin{matrix} \vec{a} & \vec{b} & \vec{c} \end{matrix}]$
$= V$
Hence this is the correct option.

Let the vectors a → ,   b → ,   c → represent three coterminous edges of a…

Solution

Let the vectors a &#8594; , &#160; b &#8594; , &#160; c &#8594; represent three coterminous edges of a…

Solution

Let the vectors a → , b → , c → represent three coterminous edges of a…