First slide

Scalar triple product of vectors

Question

$The altitude of a parallelepiped whose three coterminous edges are vectors$ $\bar{a} = \bar{i} + \bar{j} + \bar{k}, \bar{b} = 2 \bar{i} + 4 \bar{j} - \bar{k} and \bar{c} = \bar{i} + \bar{j} + 3 \bar{k} with \bar{a} and \bar{b} as sides of the base of the$ $parallelopiped is$

Moderate

A

$\frac{4}{\sqrt{19}}$
B

$\frac{2}{\sqrt{19}}$
C

$\frac{2 \sqrt{38}}{19}$
D

$\frac{1}{\sqrt{19}}$

Solution

$L e t \overset{}{\vec{a} = \hat{i} + \hat{j} + \hat{k}} \vec{b} = 2 \hat{i} + 4 \hat{j} - \hat{k} \vec{c} = \hat{i} + \hat{j} + 3 \hat{k}$
$v o l u m e o f p a r a l l e l o p i p e d i s [\bar{a} \bar{b} \bar{c}] = |\begin{matrix} 1 & 1 & 1 \\ 2 & 4 & - 1 \\ 1 & 1 & 3 \end{matrix}| = 4$
$b a s e a r e a w h o s e s i d e s a r e \bar{a}, \bar{b} i s = |a \times b| = ||\begin{matrix} i & j & k \\ 1 & 1 & 1 \\ 2 & 4 & - 1 \end{matrix}|| = \sqrt{38}$
height of parallelopiped= $\frac{v o l u m e}{b a s e a r e a} = \frac{4}{\sqrt{38}}$ =

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