Question

A vector $\vec{a}$ makes equal angles with all the three axes. If the magnitude of the vector is $5\sqrt{3}$ units, then find $\vec{a}$.

• A. $\vec{a} = 5\hat{i} + 5\hat{j} + 5\hat{k}$
• B. $\vec{a} = -5\hat{i} - 5\hat{j} - 5\hat{k}$
• C. $\vec{a} = 5\hat{i} - 5\hat{j} + 5\hat{k}$
• D. $\vec{a} = -5\hat{i} + 5\hat{j} - 5\hat{k}$

Answer 1

Answer: (A), (B)

The vector $\vec{a}$ can be $5\hat{i} + 5\hat{j} + 5\hat{k}$ or $-5\hat{i} - 5\hat{j} - 5\hat{k}$.

Answer 2

A vector making equal angles with all three axes has direction cosines $l=m=n$. The relation $l^2+m^2+n^2=1$ implies $3l^2=1$, so $l = \pm \frac{1}{\sqrt{3}}$. The components are $a_x = |\vec{a}|l$, $a_y = |\vec{a}|m$, $a_z = |\vec{a}|n$. With $|\vec{a}| = 5\sqrt{3}$, the components are $\pm 5$. Thus, $\vec{a} = \pm 5\hat{i} \pm 5\hat{j} \pm 5\hat{k}$, with the signs being the same for all components.