Question

The vector $\hat{i}+\hat{j}+\hat{k}$ is

• A. Equally inclined with the coordinate axes.
• B. inclined more towards the x-y plane.
• C. inclined more towards the x-z plane.
• D. inclined more towards the y-z plane.

Answer 1

The correct option is A Equally inclined with the coordinate axes.

If $\alpha$ is the angle made by the vector with x-axis and x-axis is represented by unit vector $\hat{i}$
$cos\alpha =\frac{(\hat{i}+\hat{j}+\hat{k}).\left(\hat{i}\right)}{|\hat{i}+\hat{j}+\hat{k}||\hat{i}|}=\frac{1}{\sqrt{3}}$

Similarly, for y-axis and z-axis,
$cos\beta =\frac{(\hat{i}+\hat{j}+\hat{k}).\left(\hat{j}\right)}{|\hat{i}+\hat{j}+\hat{k}||\hat{j}|}=\frac{1}{\sqrt{3}}$

$cos\gamma =\frac{(\hat{i}+\hat{j}+\hat{k}).\left(\hat{k}\right)}{|\hat{i}+\hat{j}+\hat{k}||\hat{k}|}=\frac{1}{\sqrt{3}}$

$\therefore \hat{i}+\hat{j}+\hat{k}$ is equally inclined to the three axes.