First slide

Vector triple product

Question

$The altitude of parallelopiped whose three coterminous edges are the vectors,$ $\vec{A} = \hat{i} + \hat{j} + \hat{k}; \vec{B} = 2 \hat{i} + 4 \hat{j} - \hat{k} and \vec{C} = \hat{i} + \hat{j} + 3 \hat{k} with \vec{A} and \vec{B} as the sides of$ $the base of the parallelopipied, is$

Moderate

A

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

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

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

None of these

Solution

$h = \frac{|[\vec{A} \vec{B} \vec{C}]|}{| \vec{A} \times \vec{B} |} a n d | \vec{A} \times \vec{B} | = \sqrt{{\vec{a}}^{2} {\vec{b}}^{2} - {(\vec{a} \cdot \vec{b})}^{2}}$
${\vec{a}}^{2} {\vec{b}}^{2} - {(\vec{a} \cdot \vec{b})}^{2} {=3 ×21-(2+4-1)}^{2} =63-25=38$
$\vec{A} × \vec{B} = - 5 \hat{i} +3 \hat{j} +2 \hat{k}$
$[\vec{A} \vec{B} \vec{C}] = (\vec{A} × \vec{B}) . \vec{C} = (- 5 \hat{i} +3 \hat{j} +2 \hat{k}) \cdot (\hat{i} + \hat{j} +3 \hat{k}) = 4$
$∴ h = \frac{4}{\sqrt{38}} = \frac{4 \sqrt{38}}{38} = \frac{2 \sqrt{38}}{19}$

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