1. **Problem Statement:** Find the length of the perpendicular drawn from point $B(1,-1)$ to the line of action of the force $\vec{F} = 4\hat{i} - 3\hat{j}$ acting at point $A(2,-1)$. 2. **Understanding the problem:** The force vector $\vec{F}$ defines a line passing through point $A$. We want the shortest distance from point $B$ to this line. 3. **Formula for distance from point to line:** If a line passes through point $A(x_1,y_1)$ with direction vector $\vec{d} = (a,b)$, the perpendicular distance $d$ from point $B(x_0,y_0)$ to the line is given by: $$ d = \frac{|(\vec{AB} \times \vec{d})|}{|\vec{d}|} $$ where $\vec{AB} = (x_0 - x_1, y_0 - y_1)$ and $\times$ denotes the 2D cross product magnitude. 4. **Calculate vectors:** $\vec{AB} = (1 - 2, -1 - (-1)) = (-1, 0)$ $\vec{d} = (4, -3)$ 5. **Calculate cross product magnitude:** $$ |\vec{AB} \times \vec{d}| = |(-1)(-3) - (0)(4)| = |3 - 0| = 3 $$ 6. **Calculate magnitude of $\vec{d}$:** $$ |\vec{d}| = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 $$ 7. **Calculate distance:** $$ d = \frac{3}{5} $$ **Final answer:** The length of the perpendicular is $\frac{3}{5}$ length units. This corresponds to option (b).