Vector Coordinates

Finding Vector Coordinates

If point A has coordinates (x₁, y₁) and point B has coordinates (x₂, y₂), then: →AB = (x₂ − x₁, y₂ − y₁).

The midpoint M of segment AB: M = ((x₁ + x₂)/2, (y₁ + y₂)/2). In 3D: →AB = (x₂ − x₁, y₂ − y₁, z₂ − z₁).

Length (Modulus) of a Vector

In 2D: |a| = √(a₁² + a₂²). In 3D: |a| = √(a₁² + a₂² + a₃²).

Example: A(1, 3) and B(4, 7): →AB = (3, 4), length = √(9+16) = 5.

Direction Cosines

In 3D: cos α = a₁/|a|, cos β = a₂/|a|, cos γ = a₃/|a|, where α, β, γ are the angles the vector makes with the coordinate axes. They satisfy: cos²α + cos²β + cos²γ = 1.

The distance between two points: d(A,B) = |→AB|.