One vertex of a rectangular parallelepiped is at the origin O and the lengths of its edges along x, y and z axes are 3, 4 and 5 units respectively. Let P be the vertex (3, 4, 5). Then the shortest distance between the diagonal OP and an edge parallel to z axis, not passing through O or P is:

Engineering

Mathematics

Shortest Distance Between Two Lines

Question

$12 \sqrt{5}$

$\frac{12}{\sqrt{5}}$

$\frac{12}{5 \sqrt{5}}$

$\frac{12}{5}$

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Equation of line CD is

$\vec{r} = 4 \hat{j} + λ 5 \hat{k}$

Equation of OP is

$\vec{r} = μ (3 \hat{i} + 4 \hat{j} + 5 \hat{k})$

S.D. = $| \frac{({\vec{a}}_{1} - {\vec{a}}_{2}) ({\vec{b}}_{1} \times {\vec{b}}_{2})}{| {\vec{b}}_{1} \times {\vec{b}}_{2} |} | = \frac{| \begin{matrix} 0 & 4 & 0 \\ 0 & 0 & 5 \\ 3 & 4 & 5 \end{matrix} |}{| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 0 & 0 & 5 \\ 3 & 4 & 5 \end{matrix} |}$

S.D, = $\frac{12}{5}$