Question

A vector of magnitude $4$ which is equally inclined to the vectors $\hat{i}+\hat{j},\hat{j}+\hat{k}$ and $\hat{k}+\hat{i}$ is

• A. $\frac{4}{\sqrt{3}}(\hat{i}-\hat{j}-\hat{k})$
• B. $\frac{4}{\sqrt{3}}(\hat{i}+\hat{j}-\hat{k})$
• C. $\frac{4}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$
• D. None of these

Answer 1

The correct option is C $\frac{4}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$

Let us assume the vector to be $a\hat{i}+b\hat{j}+c\hat{k}$
This is equally inclined to $\hat{i}+\hat{j},\hat{j}+\hat{k},\hat{i}+\hat{k}$
Thus, $a+b=b+c=a+c$ using the dot product condition for equal inclination.
$\Rightarrow a=b=c$
Now since a vector of magnitude $4$ is required, we get the answer as $\frac{4}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$