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

To solve the problem, we are given that a vector \( \mathbf{a} \) makes equal angles with all three coordinate axes and has a magnitude of \( 5\sqrt{3} \) units. We need to find the vector \( \mathbf{a} \).

1. Direction Cosines of the Vector:
If a vector makes equal angles with the x-, y-, and z-axes, then its direction cosines are equal. Let the common direction cosine be \( l \).

Since the sum of the squares of direction cosines equals 1:
\[ l^2 + l^2 + l^2 = 1 \Rightarrow 3l^2 = 1 \Rightarrow l^2 = \frac{1}{3} \Rightarrow l = \frac{1}{\sqrt{3}} \]

2. Unit Vector in Direction of \( \mathbf{a} \):
The unit vector making equal angles with the axes is:
\[ \hat{\mathbf{a}} = \left\langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right\rangle \]

3. Multiply by Magnitude:
We now multiply the unit vector by the magnitude \( 5\sqrt{3} \):
\[ \mathbf{a} = 5\sqrt{3} \cdot \left\langle \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right\rangle = \left\langle 5, 5, 5 \right\rangle \]

Final Answer:
The vector \( \mathbf{a} \) is \( \boxed{\langle 5, 5, 5 \rangle} \).