[Math] shortest distance between two vectors

vectors

Whilst working on vectors I have come across a lot of problems like this. I am able to work it out for the shortest distance from a vector to a point, but not from a vector to a vector. Here is my usual method for a question asking the shortest distance from a vector (passing through $A$ and $B$) to a point $C$:
$$\vec{OA}=a$$
$$\vec{OB}=b$$
$$\vec{OC}=c$$
Where $O$ is the origin. We know that the equation for a line passing through $A$ and $B$ is:
$$\vec{r}=\mu(b-a)+a$$
we also know that at the closest distance a line from $C$ to $\vec{r}$ is perpendicular to $\vec{r}$.
I would now define:
$$\vec{r}=\begin{pmatrix}\mu(b_1-a_1)+a_1\\\mu(b_2-a_2)+a_2\\\mu(b_3-a_3)+a_3\end{pmatrix}=\begin{pmatrix}d_1\\d_2\\d_3\end{pmatrix}$$
so the distance from $\vec{r}$ to $C$ is:
$$l=\sqrt{(d_1-c_1)^2+(d_2-c_2)^2+(d_3-c_3)^2}$$
now find the point at which $\frac{dl}{d\mu}=0$ solving for $\mu$ and subbing into the equation.

However, I am aware that there are much easier methods for find the point $N$ and the shortest distance $|\vec{NC}|$ involving the fact that $\vec{r}\bullet\vec{NC}=0$ or potentially cross product as well. Does anyone have a tutorial for this method? Also, how would I solve this same problem but finding the minimum distance between two vectors?
Thanks

Best Answer

For problems like this one you don't need derivatives.

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Suppose that you know the coordinates of points $A(x_A, y_A, z_A)$, $B(x_B, y_B, z_B)$ and components of vectors $\vec a=(a_x,a_y,a_z)$, $\vec b=(b_x,b_y,b_z)$. The shortest distance between lines is represented with segment $CD$ and that segment is prependicular both to $\vec a$ and $\vec b$.

Now you have:

$$AC=\mu \vec a$$

$$BD=\lambda \vec b$$

$$\vec {CD} \bot \vec a \implies \vec{CD}\cdot \vec a=0$$

$$\vec {CD} \bot \vec b \implies \vec{CD}\cdot \vec b=0$$

...or, in scalar form:

$$x_C-x_A=\lambda a_x$$

$$y_C-y_A=\lambda a_y$$

$$z_C-z_A=\lambda a_z$$

$$x_D-x_B=\mu b_x$$

$$y_D-y_B=\mu b_y$$

$$z_D-z_B=\mu b_z$$

$$(x_D-x_C)a_x+(y_D-y_C)a_y+(z_D-z_C)a_z=0$$

$$(x_D-x_C)b_x+(y_D-y_C)b_y+(z_D-z_C)b_z=0$$

You have 8 linear equations and 8 unknowns: $x_C, y_C, z_C, x_D, y_D, z_D, \lambda, \mu$:

  • From the first three equations express $x_C, y_C, z_C$ in terms of $\lambda$.
  • From the next three equations express $x_D, y_D, z_D$ in terms of $\mu$.
  • Replace all that into the last two equations and you have a system of two equations with two unknowns $(\lambda,\mu)$.
  • Solve, find coordinates of points $C,D$
  • Calculate distance CD.
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