If a, b, c, p, q and r \in R – {0} such that ap + bq + cr + \sqrt{\left(\right. a^{2} + b^{2} + c^{2} \left.\right) \left(\right. p^{2} + q^{2} + r^{2} \left.\right)} = 0 then find the value of \frac{aq}{bp} + \frac{br}{cq} + \frac{cp}{ar} .

Engineering

Mathematics

Scalar Product of Two Vectors

Question

If a, b, c, p, q and r $\in$ R – {0} such that ap + bq + cr + $\sqrt{(a^{2} + b^{2} + c^{2}) (p^{2} + q^{2} + r^{2})}$ = 0 then find the value of $\frac{aq}{bp} + \frac{br}{cq} + \frac{cp}{ar} .$

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Let $\vec{m} = a \hat{i} + b \hat{j} + c \hat{k} and \vec{n} = p \hat{i} + q \hat{i} + r \hat{k}$

$∴ \vec{m} \cdot \vec{n} = - | \vec{m} | | \vec{n} |$

$∴ \vec{m} is antiparallel to \vec{n}$

$∴ \frac{a}{p} = \frac{b}{q} = \frac{c}{r}$

$∴ \frac{aq}{bp} + \frac{br}{cq} + \frac{cp}{ar} = 3$