Question

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

Hint:

A vector making equal angles with all axes has direction cosines \( \frac{1}{\sqrt{3}} \). Multiply by the magnitude to find the vector components.

Answer 1

Step 1: Define the unit direction vector.
Since \( \mathbf{a} \) makes equal angles with the coordinate axes, let the direction cosines be \( l, m, n \). For a vector making equal angles with all three axes: \[ l = m = n. \] Since the sum of the squares of the direction cosines is always 1, we write: \[ l^2 + m^2 + n^2 = 1. \] Substituting \( l = m = n \): \[ 3l^2 = 1. \] \[ l^2 = \frac{1}{3}, \quad l = \frac{1}{\sqrt{3}}. \] Thus, the direction cosines are:
\[ \left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}\right). \] Step 2: Find the components of \( \mathbf{a} \).
The vector \( \mathbf{a} \) is given by: \[ \mathbf{a} = |\mathbf{a}| \times (\text{direction cosines}). \] Given that \( |\mathbf{a}| = 5\sqrt{3} \), we compute: \[ \mathbf{a} = 5\sqrt{3} \left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right). \] \[ \mathbf{a} = (5,5,5). \] Final Answer:
\[ \mathbf{a} = 5\hat{i} + 5\hat{j} + 5\hat{k}. \]