Let PR→=3i^+j^−2k^ and SQ→=i^−3j^−4k^ determine diagonals of parallelogram PQRS and PT→=i^+2j^+3k^ be another vector. Then the volume of the parallelepiped determined by the vectors PT→,PQ→ and PS→ is

PR→=PQ→+PS→ SQ→=PQ→−PS→ PS→=PR→−SQ→2 PQ→=PR→−SQ→2 v=|[PQ→PS→PT→]| v=14|[PR→+SQ→, PR→−SQ→,PT→]| v=12|[PR→,SQ→,PT→]| 12|31−21−3−4123|=12(−3−7−10)=10

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Let $\vec{P R} = 3 \hat{i} + \hat{j} - 2 \hat{k}$ and $\vec{S Q} = \hat{i} - 3 \hat{j} - 4 \hat{k}$ determine diagonals of parallelogram $P Q R S$ and $\vec{P T} = \hat{i} + 2 \hat{j} + 3 \hat{k}$ be another vector. Then the volume of the parallelepiped determined by the vectors $\vec{P T}, \vec{P Q}$ and $\vec{P S}$ is

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answer is 10.

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Detailed Solution

$\vec{P R} = \vec{P Q} + \vec{P S}$
$\vec{S Q} = \vec{P Q} - \vec{P S}$
$\vec{P S} = \frac{\vec{P R} - \vec{S Q}}{2}$
$\vec{P Q} = \frac{\vec{P R} - \vec{S Q}}{2}$
$v = | [\begin{matrix} \vec{P Q} & \vec{P S} & \vec{P T} \end{matrix}] |$
$v = \frac{1}{4} | [\vec{P R} + \vec{S Q}, \vec{P R} - \vec{S Q}, \vec{P T}] |$
$v = \frac{1}{2} | [\vec{P R}, \vec{S Q}, \vec{P T}] |$
$\frac{1}{2} | \begin{matrix} 3 & 1 & - 2 \\ 1 & - 3 & - 4 \\ 1 & 2 & 3 \end{matrix} | = \frac{1}{2} (- 3 - 7 - 10) = 10$

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