Out of the following vectors, identify the one which is normal to the force, $\vec{F} = 4\hat{i} - 3\hat{j}$.

Two vectors are normal to each other if their dot product is zero.

Given force vector $\vec{F} = 4\hat{i} - 3\hat{j}$.

Checking the dot product of $\vec{F}$ with each of the given vectors:

For option (1): $(4\hat{i} - 3\hat{j}) \cdot (-6\hat{i}) = (4)(-6) + (-3)(0) = -24 \neq 0$

For option (2): $(4\hat{i} - 3\hat{j}) \cdot (7\hat{i}) = (4)(7) + (-3)(0) = 28 \neq 0$

For option (3): $(4\hat{i} - 3\hat{j}) \cdot (3\hat{i} + 4\hat{j}) = (4)(3) + (-3)(4) = 12 - 12 = 0$

For option (4): $(4\hat{i} - 3\hat{j}) \cdot (4\hat{i} + 3\hat{j}) = (4)(4) + (-3)(3) = 16 - 9 = 7 \neq 0$

Since the dot product is zero for the vector $3\hat{i} + 4\hat{j}$, it is normal to the given force vector.