Vectors

A vector has size and direction, like a journey of “3 right and 2 up”. Vectors are added nose to tail, and multiplying by a number scales them. They let you prove things about shapes, like lines being parallel.

Key facts

  • Column vector (3, 2) means 3 right, 2 up.
  • Add nose to tail: AB + BC = AC.
  • Parallel vectors are multiples of each other.
  • Magnitude of (x, y) = √(x² + y²).

Worked examples

Example 1

a = (2, 3) and b = (−1, 4). Find 2a − b.

  1. 2a = (4, 6)
  2. (4, 6) − (−1, 4) = (5, 2)

Answer: (5, 2)

Example 2

OA = a, OB = b, and M is the midpoint of AB. Find OM in terms of a and b.

  1. AB = b − a
  2. OM = OA + ½AB = a + ½(b − a)
  3. = ½a + ½b

Answer: ½a + ½b