Question
For three unit vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ satisfying
$|\overrightarrow{a} - \overrightarrow{b}|^{2} + |\overrightarrow{b} - \overrightarrow{c}|^{2} + |\overrightarrow{c} - \overrightarrow{a}|^{2} = 9$ and
$|2\overrightarrow{a} + k\overrightarrow{b} + k\overrightarrow{c}| = 3$,
the positive value of k is :
Options
Solution
$|\overrightarrow{a} - \overrightarrow{b}|^{2} + |\overrightarrow{b} - \overrightarrow{c}|^{2} + |\overrightarrow{c} - \overrightarrow{a}|^{2} = 9$
$${\Rightarrow \overrightarrow{a} \cdot \overrightarrow{b} + \overrightarrow{b} \cdot \overrightarrow{c} + \overrightarrow{c} \cdot \overrightarrow{a} - = - \frac{3}{2} }{\Rightarrow \overrightarrow{a} + \overrightarrow{b} + \overrightarrow{c} = 0 \Rightarrow \overrightarrow{b} + \overrightarrow{c} = - \overrightarrow{a} }{|2\overrightarrow{a} + k(\overrightarrow{b} + \overrightarrow{c})| = 3 }{|\overrightarrow{a}(2 - k)| = 3 }$$$K = 5$ or -1
Positive value of k is 5
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