Give that vectors a , b , c . form a triangle such that a = b + c… - Vidyadip

Question

Give that vectors $\overrightarrow{\text{a}}, \overrightarrow{\text{b}}, \overrightarrow{\text{c}}.$ form a triangle such that $\overrightarrow{\text{a}} = \overrightarrow{\text{b}} + \overrightarrow{\text{c}}.$ Find p, q, r, s such that area of triangle is $5\sqrt{6}$ where $\overrightarrow{\text{a}} = \text{p }\hat{i} + \text{q }\hat{j} + \text{r }\hat{k} = \overrightarrow{\text{b}} = \text{s }\hat{i} + \text{3 }\hat{j} + \text{4 }\hat{k} \text{ and} \overrightarrow{\text{c}} = \text{3 }\hat{i} + \hat{j} - \text{2}\hat{k}.$

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Answer

$\overrightarrow{a} = \overrightarrow{b} +\overrightarrow{c} \Rightarrow p\hat{i} + q\hat{j} + r\hat{k} = (s + 3) \hat{i} + 4\hat{j} + 2\hat{k}$
$p = s + 3, q = 4, r = 2$
$\text{area} = \frac{1}{2} |\overrightarrow{b} \times\overrightarrow{c}| = 5\sqrt{6}$
$\overrightarrow{b}\times\overrightarrow{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ s & 3 & 4 \\ 3 & 1 & -2 \end{vmatrix} = -10\hat{i} + (2s + 12)\hat{j} + (s - 9) \hat{k} $
$\therefore 100 +(2s + 12)^{2} +(s - 9)^{2} = (10\sqrt{6})^{2} = 600$
$\Rightarrow s^{2} + 6s + 55 = 0\Rightarrow s = -11, p = -8, \text{or } s = 5, p = 8$

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