Solution

Vector Projection Problem

Given: A vector of magnitude 5 makes equal angles with x, y, and z axes.
To Find: Sum of magnitudes of projections on each axis.

Let angle with each axis be \( \alpha \). Then, from direction cosine identity: \[ \cos^2\alpha + \cos^2\alpha + \cos^2\alpha = 1 \Rightarrow 3\cos^2\alpha = 1 \Rightarrow \cos\alpha = \frac{1}{\sqrt{3}} \]

Projection on each axis: \( 5 \cdot \frac{1}{\sqrt{3}} \)
Sum = \( 3 \cdot \frac{5}{\sqrt{3}} = \frac{15}{\sqrt{3}} = \boxed{5\sqrt{3}} \)

✅ Final Answer: \( \boxed{5\sqrt{3}} \)