Vectors in the Coordinate Plane

Writing a vector in component form by subtracting the start point from the end point, reading the angled-bracket notation, and finding the length with Pythagoras - with worked examples on negative coordinates.

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Given a start point and an end point, two things follow immediately: the vector's component form and its length. Both come from the same pair of subtractions.

Concept What component form means

A vector in component form is one whose tail has been moved to the origin — without changing its direction or its length.

is the start point and is the end point.

Always subtract end minus start, in that order. Reversing it gives a vector pointing the opposite way.

Note Two kinds of brackets
Round brackets mark a point: is a location.
Angled brackets mark a vector: is a displacement.

The notation is doing real work here — seeing tells you at once that a vector is meant, not a position.

Example Finding the component form
A(−4, 2) B(3, −5) O ⟨7, −7⟩
The dashed arrow from the origin is the same vector as the solid one — same length, same direction, just moved.
From to :
x-component:
y-component:
⟹ ⟨7, −7⟩

Watch the double negative: subtracting is the same as adding 4.

Theorem The length of a vector
Δx Δy |AB|
The two components form the legs of a right triangle, and the vector itself is the hypotenuse — so Pythagoras gives the length.

Squaring removes any minus signs automatically, which is exactly right — a length is never negative.

Example Two lengths worked out
From to , the components are 7 and −7:
From to :
x-component: , y-component:
⟹ about 9.9 and 14.2
Summary
  1. Component form moves the vector's tail to the origin, unchanged otherwise.
  2. ⟨x, y⟩ = ⟨x₂ − x₁, y₂ − y₁⟩, always end minus start.
  3. Round brackets mark points; angled brackets mark vectors.
  4. Length comes from Pythagoras: √(Δx² + Δy²).
  5. Squaring clears the signs, so a length is always positive.
283. Relations and Functions
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