# The Value of [ → a − → B , → B − → C , → C − → a ] , Where - Mathematics

MCQ
Sum

The value of $\left[ \vec{a} - \vec{b} , \vec{b} - \vec{c} , \vec{c} - \vec{a} \right], \text { where } \left| \vec{a} \right| = 1, \left| \vec{b} \right| = 5, \left| \vec{c} \right| = 3, \text { is }$

#### Options

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#### Solution

We have

$\left[ \vec{a} - \vec{b} , \vec{b} - \vec{c} , \vec{c} - \vec{a} \right]$

$= \left( \left( \vec{a} - \vec{b} \right) \times \left( \vec{b} - \vec{c} \right) \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \left( \text { By definition of scalar triple product } \right)$

$= \left( \left( \vec{a} - \vec{b} \right) \times \vec{b} - \left( \vec{a} - \vec{b} \right) \times \vec{c} \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em}$

$= \left( \vec{a} \times \vec{b} - \vec{b} \times \vec{b} - \vec{a} \times \vec{c} + \vec{b} \times \vec{c} \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em}$

$= \left( \vec{a} \times \vec{b} - 0 - \vec{a} \times \vec{c} + \vec{b} \times \vec{c} \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em}$

$= \left( \vec{a} \times \vec{b} \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} - \left( \vec{a} \times \vec{c} \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em} + \left( \vec{b} \times \vec{c} \right) . \left( \vec{c} - \vec{a} \right) \hspace{0.167em}$

$= \left( \vec{a} \times \vec{b} \right) . \vec{c} - \left( \vec{a} \times \vec{b} \right) . \vec{a} - \left( \vec{a} \times \vec{c} \right) . \vec{c} + \left( \vec{a} \times \vec{c} \right) . \vec{a} + \left( \vec{b} \times \vec{c} \right) . \vec{c} - \left( \vec{b} \times \vec{c} \right) . \vec{a}$

$= \left[ \vec{a} \vec{b} \vec{c} \right] - \left[ \vec{a} \vec{b} \vec{a} \right] - \left[ \vec{a} \vec{c} \vec{c} \right] + \left[ \vec{a} \vec{c} \vec{a} \right] + \left[ \vec{b} \vec{c} \vec{c} \right] - \left[ \vec{b} \vec{c} \vec{a} \right]$

$= \left[ \vec{a} \vec{b} \vec{c} \right] - 0 - 0 + 0 + 0 - \left[ \vec{a} \vec{b} \vec{c} \right] \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \left( \because \left[ \vec{a} \vec{b} \vec{c} \right] = \left[ \vec{b} \vec{c} \vec{a} \right] = \left[ \vec{c} \vec{a} \vec{b} \right] \right)$

$= 0 \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em} \hspace{0.167em}$

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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 26 Scalar Triple Product
MCQ | Q 2 | Page 18