What is this formula about (position vectors)?

Question

I know this is about position vectors, but don't know exactly what its purpose is. It goes like this:-

$$\frac{(m\vec u + n\vec v)}{(m+n)}$$

Has anyone seen something similar to this?

If there's any problem in my question please inform me. Thanks!

Answer 1

Let $$\lambda = \frac{n}{m + n}$$ Then $$1 - \lambda = 1 - \frac{n}{m + n} = \frac{m + n - n}{m + n} = \frac{m}{m + n}$$ and $$\frac{m \vec{u} + n \vec{v}}{m + n} = \frac{m}{m + n}\vec{u} + \frac{n}{m + n}\vec{v} = (1 - \lambda) \vec{u} + \lambda \vec{v} = \vec{u} + \lambda (\vec{v} - \vec{u})$$ which is simply linear interpolation between $\vec{u}$ and $\vec{v}$.

In other words, the expression $$\frac{m\vec{u} + n\vec{u}}{m + n}$$ is linear interpolation from $\vec{u}$ (when $n=0$) to $\vec{v}$ (when $m=0$). Do note that $m = n = 0$ is undefined, because the denominator is zero, so at least one of them needs to be nonzero.

Geometrically, the point $\vec{p} = \frac{m\vec{u}+n\vec{v}}{m+n}$ is on the line between $\vec{u}$ and $\vec{v}$ such that $$\frac{\left\lVert \vec{p} - \vec{u} \right\rVert}{\left\lVert \vec{v} - \vec{u}\right\rVert} = \frac{m}{m + n} \quad \text{and} \quad \frac{\left\lVert \vec{p} - \vec{v} \right\rVert}{\left\lVert \vec{v} - \vec{u}\right\rVert} = \frac{n}{m + n}$$

Answer 2

What the formula describes is a point on the line which goes through the points $\vec u$ and $\vec v$. Any choice of values for $m$ and $n$, as long as $m+n\neq0$, gives such a point.

If $m$ and $n$ have the same sign, then the point is between $\vec u$ and $\vec v$.

Answer 3

Arthur's answer is quite correct. Another way to interpret the expression is that it is a weighted average of the two points. Depending on the relative size of $m$ and $n$, you're weighting $\vec{u}$ or $\vec{v}$ more, respectively.

[EDIT]: As soktinpk mentioned in the comment to this answer:

When interpreting $\vec{u}$ and $\vec{v}$ as position vectors, the weighted average is the center of mass of two objects located at $\vec{u}$ and $\vec{v}$ with a mass of $m$ and $n$ respectively.