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Pure · Vectors

Chapter 1 · 4

The idea

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.

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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

Tying points to the origin

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.

Destination minus start

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=b−a\vec{AB} = \vec{AO} + \vec{OB} = \mathbf{b} - \mathbf{a}

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