Column vectors and magnitude
A vector has both size and direction. In two dimensions it is written as a column vector (x, y), where x is the horizontal component and y the vertical component. Its magnitude (length) is found using Pythagoras.
Key idea
For a = (x, y), the magnitude is |a| = √(x2 + y2). A unit vector in the direction of a is (1⁄|a|) a, which has magnitude 1.
Operations and position vectors
Vectors are added or subtracted component by component, and a scalar k multiplies each component. The position vector of a point A is OA, measured from the origin O. The vector joining A to B is AB = OB − OA = b − a.
Two vectors are parallel when one is a scalar multiple of the other. A velocity written as a vector has a speed equal to its magnitude.
Worked example
If OA = (1, 2) and OB = (5, 7), then AB = b − a = (5 − 1, 7 − 2) = (4, 5). Its magnitude is |AB| = √(42 + 52) = √41.
Remember
- AB = b − a, not a − b.
- A unit vector always has magnitude 1.
- Parallel vectors have proportional components.