First slide

Dot product or scalar product of vectors

Question

If $\hat{a}, \hat{b} and \hat{c}$ are three unit vectors inclined to each other at an angle $θ$ , then the maximum value of $θ$ is

Moderate

A

$\frac{2 π}{3}$
B

$\frac{5 π}{3}$
C

$\frac{π}{3}$
D

$\frac{π}{2}$

Solution

$\begin{array}{l} (\hat{a} + \hat{b} + \hat{c})^{2} \geq 0 \\ \Rightarrow 3 + 2 (\hat{a} \cdot \hat{b} + \vec{b} \cdot \hat{c} + \vec{c} \cdot \vec{a}) \geq 0 \\ \Rightarrow 3 + 6 \cos θ \geq 0 \\ \Rightarrow \cos θ \geq - \frac{1}{2} \\ \Rightarrow θ = \frac{2 π}{3} \end{array}$

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