Question

A vector $\overrightarrow{A}$ has magnitude A and $\hat{A}$ is unit vector in the direction of $\overrightarrow{A}$, then which of the following are correct

• A. $\overrightarrow{A}.\hat{A}=A$
• B. $\hat{A}=\frac{\overrightarrow{A}}{A}$
• C. $\overrightarrow{A}.\overrightarrow{A}=A^2$
• D. All of the above

Answer 1

The correct option is D All of the above

$\text{Step 1: Analysing option A}$

$\overrightarrow{A}\to$vector; $|\overrightarrow{A}|=A\to$ magnitude; $\hat{A}\to$ Unit vector;

We know, $\hat{A}=\frac{\overrightarrow{A}}{|\overrightarrow{A}|}$

So, $\overrightarrow{A}.\hat{A}=\overrightarrow{A}.\frac{\overrightarrow{A}}{A}=\frac{A^2}{A}=A$

Hence option $A$ is correct.

$\text{Step 2: Analysing option B}$

$\hat{A}=\frac{\overrightarrow{A}}{A}$ [Unit Vector]

Hence option $B$ is also correct.

$\text{Step 3: Analysing option C}$

$\overrightarrow{A}.\overrightarrow{A}=A^2$

Hence option $C$ is also correct.

All the options are correct.

$\therefore$ Option $D$ is the correct answer.