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Pure · Vectors

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Position vectors

Pinning points to the origin with position vectors, the "destination minus start" rule AB = b − a, the distance between two points as |b − a|, the midpoint as ½(a + b), and finding a point that divides a segment in a ratio.

Pure · Vectors

Position vectors

Pinning points to the origin with position vectors, the "destination minus start" rule AB = b − a, the distance between two points as |b − a|, the midpoint as ½(a + b), and finding a point that divides a segment in a ratio.

Why it works

A vector floats free — it has no fixed home. To describe a point, we tie it to the origin OO. The position vector of a point AA is the vector OA⃗\vec{OA} from the origin to AA, usually called a\mathbf{a}. A point with position vector xi+yjx\mathbf{i} + y\mathbf{j} sits at the coordinates (x,y)(x, y) — position vectors and coordinates are two names for the same information.

Everything else falls out of one route through the origin. To get from AA to BB, go A→O→BA \to O \to B: AB⃗=AO⃗+OB⃗=−a+b=b−a.\vec{AB} = \vec{AO} + \vec{OB} = -\mathbf{a} + \mathbf{b} = \mathbf{b} - \mathbf{a}.Destination minus start. AB⃗=b−a\vec{AB} = \mathbf{b} - \mathbf{a} — the position vector you are heading to, minus the one you start from. Get them the wrong way round and you get BA⃗\vec{BA}, which points backwards.abb − aOABFrom that single rule:
  • Distance. The length ABAB is the magnitude of that vector,
∣AB⃗∣=∣b−a∣|\vec{AB}| = |\mathbf{b} - \mathbf{a}| — and as ever, a length is left as an exact surd.
  • Midpoint. The midpoint MM of ABAB is halfway along, so its position vector is
the average of the ends: OM⃗=12(a+b)\vec{OM} = \tfrac{1}{2}(\mathbf{a} + \mathbf{b}). (Add and halve — exactly the coordinate midpoint, in vector clothing.)
  • A point dividing ABAB in a ratio. To reach the point PP with AP:PB=m:nAP:PB = m:n,
start at AA and travel the fraction mm+n\dfrac{m}{m+n} of the way along AB⃗\vec{AB}: p=a+mm+n(b−a).\mathbf{p} = \mathbf{a} + \frac{m}{m + n}(\mathbf{b} - \mathbf{a}). The denominator is the sum m+nm + n of both parts of the ratio, not nn.