If $$e1^$$, $$e2^$$ and $$e3^$$ are three unit vectors such that $$e3^=e1^+$$ $$e2^$$ , Then the value of $$3e1^$$.$$e3^+2e1^$$. $$e2^$$ is

From the vector triangle, $$3e1^$$.$$e3^+2e1^$$. $$e2^=3.1.1$$.$$cos60o+2.1.1$$.$$cos120o$$ $$=0.5$$

Vectors

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Question

If $\hat{e_{1}}, \hat{e_{2}} and \hat{e_{3}}$ are three unit vectors such that $\hat{e_{3}} = \hat{e_{1}} + \hat{e_{2}}$ , Then the value of $(3 \hat{e_{1}} . \hat{e_{3}} + 2 \hat{e_{1}} . \hat{e_{2}})$ is

Moderate

Solution

From the vector triangle,
$\begin{array}{l} 3 \hat{e_{1}} . \hat{e_{3}} + 2 \hat{e_{1}} . \hat{e_{2}} = 3 .1.1. \cos 60^{o} + 2 .1.1. \cos 120^{o} \\ = 0 .5 \end{array}$

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