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.
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: %20%7D)
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
The dashed arrow from the origin is the same vector as the solid one — same length, same direction, just moved.
From %20%7D)
to %20%7D)
:
x-component: %20%3D%203%20%2B%204%20%3D%207%20%7D)
y-component: 
⟹ ⟨7, −7⟩
Watch the double negative: subtracting is the same as adding 4.
Theorem
The length of a vector
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 %20%7D)
to %20%7D)
:
x-component: 
, y-component: 
⟹ about 9.9 and 14.2
Summary
- Component form moves the vector's tail to the origin, unchanged otherwise.
- ⟨x, y⟩ = ⟨x₂ − x₁, y₂ − y₁⟩, always end minus start.
- Round brackets mark points; angled brackets mark vectors.
- Length comes from Pythagoras: √(Δx² + Δy²).
- Squaring clears the signs, so a length is always positive.