Key Idea: A vector is a direction plus a length. This topic writes vectors in components, finds their length, and joins points. Vectors are on Paper 1 by hand and Paper 2 with the GDC.
Vectors as components & magnitude
- Column (x, y, z)ᵀ and xi + yj + zk mean the same thing.
- |v| = √(x² + y² + z²): square, add, square-root.
- A length is never negative.
- Missing component: square both sides and keep ± at the end.
Unit & position vectors
- Unit vector v̂ = v/|v|: same direction, length 1.
- Position vector OA = the coordinates of A.
- AB = OB − OA (finish minus start). Distance = |AB|.
- Length k in a direction: unit vector × k.
Find AB first, then its length gives the distance.
(k, 12)ᵀ has magnitude 13. Find k. k² + 144 = 169, so k = ±5.
Vector of length 15 in the direction of (2, −1, 2)ᵀ? |d| = 3, so (10, −5, 10)ᵀ.
You wrote OA − OB. What did you find? BA, the opposite direction.
How do you check a unit vector? Its length must be 1.