Question

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

• A. 30°
• B. 45°
• C. 60°
• D. 90°

Answer 1

Answer: (B)

45°

Answer 2

$\cos\theta = \frac{\overset{\rightarrow}{F_{1}}.\overset{\rightarrow}{F_{2}}}{|F_{1}||F_{2}|}$

$\mathbf{=}\frac{\mathbf{(5}\widehat{\mathbf{i}}\mathbf{+ 10}\widehat{\mathbf{j}}\mathbf{- 20}\widehat{\mathbf{k}}\mathbf{).(10}\widehat{\mathbf{i}}\mathbf{- 5}\widehat{\mathbf{j}}\mathbf{- 15}\widehat{\mathbf{k}}\mathbf{)}}{\sqrt{\mathbf{25 + 100 + 400}}\sqrt{\mathbf{100 + 25 + 225}}}\mathbf{=}\frac{\mathbf{50}\mathbf{-}\mathbf{50 + 300}}{\sqrt{\mathbf{525}}\sqrt{\mathbf{350}}}$

⇒ $\mathbf{\cos}\mathbf{\theta}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{2}}}\mathbf{\therefore}$ $\mathbf{\theta = 45}\mathbf{{^\circ}}$