First slide

Scalars and Vectors

Question

Two forces ${\vec{F}}_{1} = 5 \hat{i} + 10 \hat{j} - 20 \hat{k} and {\vec{F}}_{2} = 10 \hat{i} - 5 \hat{j} - 15 \hat{k}$ act on a single point. The angle between ${\vec{F}}_{1} and {\vec{F}}_{2}$ is nearly

Moderate

A

$\cos^{- 1} (\frac{12}{7} \sqrt{6})$
B

$\cos^{- 1} (\frac{2}{7} \sqrt{6})$
C

$\cos^{- 1} (\frac{2}{7 \sqrt{6}})$
D

$\cos^{- 1} (\frac{7}{2 \sqrt{6}})$

Solution

$\begin{array}{l} \cos θ = \frac{\vec{F_{1}} \cdot \vec{F_{2}}}{|F_{1} ∥ F_{2}|} \\ = \frac{(5 \hat{i} + 10 \hat{j} - 20 \hat{k}) \cdot (10 \hat{i} - 5 \hat{j} - 15 \hat{k})}{\sqrt{25 + 100 + 400} \sqrt{100 + 25 + 225}} = \frac{50 - 50 + 300}{\sqrt{525} \sqrt{350}} = \frac{12}{7 \sqrt{6}} \\ \Rightarrow \cos θ = \frac{2}{7} \sqrt{6} \\ ∴ θ = \cos^{- 1} (\frac{2}{7} \sqrt{6}) \end{array}$

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