Let \overset{\rightarrow}{PR} = 3 \hat{i} + \hat{j} - 2 \hat{k} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \overset{\rightarrow}{SQ} \textrm{ } = \hat{i} - 3 \hat{j} - 4 \hat{k} determine diagonals of a parallelogram PQRS and \overset{\rightarrow}{PT} = \hat{i} + 2 \hat{j} + 3 \hat{k} be another vector. Then the volume of the parallelepiped determined by the vectors \overset{\rightarrow}{PT} , \textrm{ } \overset{\rightarrow}{PQ} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \overset{\rightarrow}{PS} is

Engineering

Mathematics

Scalar Triple Product

Question

Let $\vec{PR} = 3 \hat{i} + \hat{j} - 2 \hat{k} and \vec{SQ} = \hat{i} - 3 \hat{j} - 4 \hat{k}$ determine diagonals of a parallelogram PQRS and $\vec{PT} = \hat{i} + 2 \hat{j} + 3 \hat{k}$ be another vector. Then the volume of the parallelepiped determined by the vectors $\vec{PT}, \vec{PQ} and \vec{PS}$ is

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$Given, \vec{a} + \vec{b} = \hat{i} + 2 \hat{j} + 3 \hat{k}$

$\vec{b} - \vec{a} = \hat{i} - 3 \hat{j} - 4 \hat{k}$

$∴ \vec{a} = \hat{i} + 2 \hat{j} + \hat{k}, \vec{b} = 2 \hat{i} - \hat{j} - 3 \hat{k}$

$and \vec{PT} = \hat{i} + 2 \hat{j} + 3 \hat{k}$

$∴ volume = [\vec{a} \vec{b} \vec{PT}] = 10$