The idea

Vector proofs

What a vector equation actually proves — a scalar multiple gives parallel and nothing more, a multiple plus a shared point gives collinear — and how to write the conclusion in the form a mark scheme rewards.

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Geometry & measures · Vectors

Vector proofs

Why it works

Only how far, and which way

A vector holds only two pieces of information: how far, and in which direction. It carries nothing at all about where.

\overrightarrow{XY}

is the journey from

X

Y

; it does not know where on the page

X

happens to sit. Everything a vector proof can and cannot do falls out of that one sentence.

Fact 1: a multiple proves parallel — only

\overrightarrow{XY} = k\,\overrightarrow{PQ}

for some number

k

, then the journey

X \to Y

runs in the same direction as

P \to Q

(or exactly the reverse, when

k

is negative) and is

k

times as long. Same direction

\Rightarrow

the two segments are parallel. That is the whole of the deduction. It says nothing whatsoever about where those segments are.

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