Question

multiple choice question
if d is the distance between two coordinates and $d_x$, $d_y$, and $d_z$ are the distance components, then the force f is given by:

$f = fd(d_x\mathbf{i}+d_y\mathbf{j}+d_z\mathbf{k})$.
$f = f(d_x\mathbf{i}+d_y\mathbf{j}+d_z\mathbf{k})$.
$f = d(d_x\mathbf{i}+d_y\mathbf{j}+d_z\mathbf{k})$.
$f = \frac{f}{d}(d_x\mathbf{i}+d_y\mathbf{j}+d_z\mathbf{k})$.

Explanation:

Step1: Recall force - vector decomposition concept

Force vector can be decomposed based on distance components. The magnitude of the force $F$ in the direction of the unit - vector along the distance vector with components $d_x$, $d_y$, $d_z$ is considered. The unit - vector along the distance vector $\vec{d}=d_x\mathbf{i} + d_y\mathbf{j}+d_z\mathbf{k}$ has magnitude $|\vec{d}|=\sqrt{d_x^{2}+d_y^{2}+d_z^{2}} = d$. The force vector $\vec{F}$ can be written as the product of the magnitude of the force $F$ and the unit - vector in the direction of the distance vector. The unit - vector in the direction of $\vec{d}$ is $\frac{1}{d}(d_x\mathbf{i} + d_y\mathbf{j}+d_z\mathbf{k})$. So, $\vec{F}=F\frac{1}{d}(d_x\mathbf{i} + d_y\mathbf{j}+d_z\mathbf{k})=\frac{F}{d}(d_x\mathbf{i} + d_y\mathbf{j}+d_z\mathbf{k})$.

Answer:

$\vec{F}=\frac{F}{d}(d_x\mathbf{i} + d_y\mathbf{j}+d_z\mathbf{k})$ (the last option in the multiple - choice list)