Question

Let $\theta$ be the angle between two unit vectors $\mathbf{\hat{a}}$ and $\mathbf{\hat{b}}$ such that $\sin \theta = \frac{3}{5}$. Then, $\mathbf{\hat{a}} \cdot \mathbf{\hat{b}}$ is equal to:

• A. $\pm \frac{3}{5}$
• B. $\pm \frac{3}{4}$
• C. $\pm \frac{4}{5}$
• D. $\pm \frac{4}{3}$

Hint:

Remember that $\sin^2 \theta + \cos^2 \theta = 1$. Use this identity to compute $\cos \theta$ when $\sin \theta$ is given.

Answer 1

The Correct Option is C

The dot product of two unit vectors is given by: $$\mathbf{\hat{a}} \cdot \mathbf{\hat{b}} = \cos \theta.$$ 

Using the Pythagorean identity: $$\sin^2 \theta + \cos^2 \theta = 1.$$ 

Substitute $\sin \theta = \frac{3}{5}$: $$\left( \frac{3}{5} \right)^2 + \cos^2 \theta = 1.$$ Simplify: $$\frac{9}{25} + \cos^2 \theta = 1 \implies \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25}.$$ 

Thus: $$\cos \theta = \pm \sqrt{\frac{16}{25}} = \pm \frac{4}{5}.$$ 

Final Answer: $\mathbf{\hat{a}} \cdot \mathbf{\hat{b}} = \pm \frac{4}{5}$, (C).