The Section Formula

sectionformula
 

The statement itself, fresh from my lecture notes!

 

A fairly simple result involving Vectors, more often than not misconstrued as something difficult to understand (like most Mathematics these days is…).

It’s super duper simple, but when it comes to math, that’s probably what throws people off the most; they expect it to be hard, but then, paradoxically, it manages to fly under their “cognitive radar” because it’s too simple.

Jeeesus Christ, I mean, I ain’t saying mathematics teachers shouldn’t dumb the material down for comprehension, but they should do it the right way.

People are weird, man.

Anyways, here’s what the formula is saying:

If we have two points, A and B, that are some distance apart, then the point lying on the line segment joining the two, as specified by the relation \frac{\vec{AP}}{\vec{PB}} = \frac{m}{n} showing the ratio of the line segment lengths connecting the point to either of the other points, is given by the formula \frac{m\vec{b}+n\vec{a}}{m+n} where those vectors “a” and “b” are the position vectors of the two original points A and B.

It’s almost too simple to memorize (you’d maybe wanna remember which variable is which and all that, just in case it’s asked in an exam setting). But otherwise, it’s easier to do some Algebra.

Remember that the vector from a point A to point B is \vec{AB}=\vec{b}-\vec{a}

So taking that first equation we’re given: \frac{\vec{AP}}{\vec{PB}} = \frac{m}{n} You just rearrange it (using that lil factoid I dished out just now) to get n(\vec{p}-\vec{a}) = m(\vec{b}-\vec{p})

Isolate that “p”, which is the position vector of that point, and you’ll get the aforementioned Section Formula!!!

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