We are given a vector \( \vec{A} \) of magnitude 5 units, making equal angles with each of the three mutually perpendicular axes (x, y, and z).
Step 1: Express the Vector Components
Since the vector makes equal angles with the coordinate axes, let the angle made with each axis be \( \theta \). The direction cosines are given by: \[ \cos \theta = \cos \alpha = \cos \beta = \cos \gamma \] where \( \alpha, \beta, \gamma \) are the angles the vector makes with the x, y, and z axes.
For a unit vector with equal angles with all axes: \[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \] Since \( \cos \alpha = \cos \beta = \cos \gamma = \cos \theta \), we substitute: \[ 3 \cos^2 \theta = 1 \] \[ \cos^2 \theta = \frac{1}{3} \] \[ \cos \theta = \frac{1}{\sqrt{3}} \] Step 2: Find Projections on Each Axis
The component of vector \( \vec{A} \) along each axis is: \[ A_x = A \cos \theta, \quad A_y = A \cos \theta, \quad A_z = A \cos \theta \] Given \( | \vec{A} | = 5 \), we get: \[ A_x = 5 \times \frac{1}{\sqrt{3}} = \frac{5}{\sqrt{3}} \] \[ A_y = \frac{5}{\sqrt{3}}, \quad A_z = \frac{5}{\sqrt{3}} \] Step 3: Compute the Sum of the Magnitudes of Projections \[ A_x + A_y + A_z = \frac{5}{\sqrt{3}} + \frac{5}{\sqrt{3}} + \frac{5}{\sqrt{3}} \] \[ = 3 \times \frac{5}{\sqrt{3}} = \frac{15}{\sqrt{3}} \] \[ = 5\sqrt{3} \] Final Answer: \[ {5\sqrt{3}} \]