First slide

Dot product or scalar product of vectors

Question

If $\hat{a}, \hat{b} and \hat{c}$ are three unit vectors and $\hat{a} + \hat{b} + \hat{c}$ is also a unit vector and $θ_{1}, θ_{2} and θ_{3}$ are angles between the vectors $\hat{a}, \hat{b}; \hat{b}, \hat{c} and \hat{c}, \hat{a}$ respectively, then among $θ_{1}, θ_{2} and θ_{3}$

Moderate

A

all are acute angles
B

all are right angles
C

at lest one is obtuse angle
D

none of these

Solution

$\begin{array}{l} | \vec{a} + \vec{b} + \vec{c} |^{2} = 1 \\ \Rightarrow | \vec{a} |^{2} + | \vec{b} |^{2} + | \vec{c} |^{2} + 2 | \vec{a} | | \vec{b} | \cos θ_{1} \\ + 2 | \vec{b} | | \vec{c} | \cos θ_{2} + 2 | \vec{c} ∥ \vec{a} | \cos θ_{3} = 1 \\ \Rightarrow \cos θ_{1} + \cos θ_{2} + \cos θ_{3} = - 1 \end{array}$
Hence, one of $θ_{1}, θ_{2} and θ_{3}$ should be an obtuse angle.

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