Column vectors
- A vector has a size and a direction. A column vector shows the movement across (top number) and up (bottom number).
- In these notes, a column vector is written across the page as (3, 2), meaning 3 right and 2 up. In the exam it is written with the 3 above the 2.
- Negative numbers mean left or down: (−4, 1) means 4 left and 1 up.
- Vectors are printed in bold (a), written with an arrow over two letters (from A to B), or underlined when handwritten.
Adding, subtracting and multiplying
- Add or subtract the top numbers, then the bottom numbers: (3, 2) + (1, −5) = (4, −3).
- Multiplying by a number (a scalar) multiplies both parts: 3 × (2, −1) = (6, −3). The result is parallel to the original vector.
- −a has the same length as a but points the opposite way.
Vector routes (Higher)
- To get from one point to another, follow a route along vectors you know: AC = AB + BC.
- If AB = a, then BA = −a. If M is the midpoint of AB, then AM = ½a.
Vector proofs (Higher)
- Two vectors are parallel if one is a multiple of the other: 6a + 3b = 3(2a + b), so it is parallel to 2a + b and three times as long.
- If AB and BC are parallel and share the point B, then A, B and C lie on one straight line (they are collinear).
Key terms
- Vector
- A quantity with both size and direction.
- Scalar
- A quantity with size only, such as a number.
- Column vector
- A vector written as a movement across and up.
- Magnitude
- The length (size) of a vector.
- Parallel vectors
- Vectors where one is a multiple of the other.
- Collinear
- Lying on the same straight line.
- Resultant
- The single vector that has the same effect as two or more vectors added together.