Question

Which of the following is a unit vector in the direction of $\hat{i}+\hat{j}+\hat{k}$.

• A. $\hat{i}+\hat{j}+\frac{k}{\sqrt{3}}$
• B. $\frac{1}{3}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\frac{1}{\sqrt{3}}\hat{k}$
• C. $\frac{\hat{i}+\hat{j}+\hat{k}}{\sqrt{2}}$
• D. None of these

Answer 1

The correct option is B $\frac{1}{3}\hat{i}+\frac{1}{\sqrt{3}}\hat{j}+\frac{1}{\sqrt{3}}\hat{k}$

Magnitude of $(\hat{i}+\hat{j}+\hat{k})=\sqrt{1^2+1^2+1^2}=\sqrt{3}$
Now from scalar multiplication of vectors, we know that if we multiply a vector with a positive scalar, its direction remains unchanged, but its magnitude changes. So if I multiply $\hat{i}+\hat{j}+\hat{k}by\frac{1}{\sqrt{3}},$ what will be its magnitude? Let’s check the magnitude of $\frac{\hat{i}}{\sqrt{3}}+\frac{\hat{j}}{\sqrt{3}}+\frac{\hat{k}}{\sqrt{3}}=\sqrt{{\left(\frac{1}{\sqrt{3}}\right)}^2+{\left(\frac{1}{\sqrt{3}}\right)}^2+{\left(\frac{1}{\sqrt{3}}\right)}^2}=1$
Now this has magnitude 1 with the direction same as that of $\hat{i}+\hat{j}+\hat{k}$
Why the directions of $\frac{\hat{i}}{\sqrt{3}}+\frac{\hat{j}}{\sqrt{3}}+\frac{\hat{k}}{\sqrt{3}}and\hat{i}+\hat{j}+\hat{k}$ are same? Because they are just scalar multiples of each other. And scalar multiplication does not change the direction of a vector but only changes its magnitude.
So $\frac{\hat{i}}{\sqrt{3}}+\frac{\hat{j}}{\sqrt{3}}+\frac{\hat{k}}{\sqrt{3}}$ is a unit vector in the direction of $\hat{i}+\hat{j}+\hat{k}$.
I have done nothing, just divided the vector by its magnitude to make its magnitude = 1, keeping the direction same as that of original vector.