Vector Geometry

Representing Vectors

Vectors can be shown in different ways:

Notation:

• When typed, vectors are written in bold letters: a, b, c
• When handwritten, vectors are underlined: a, b, c (with a line underneath)
• If a vector goes from point A to point B, we write it as AB with an arrow on top: AB→

Visual representation: A vector is drawn as an arrow:

• The length of the arrow shows the magnitude (size)
• The direction of the arrow shows which way the vector points

Column Vector Form

The most common way to write vectors in calculations is as a column vector. This shows the movement in two directions:

$\begin{pmatrix} x \\ y \end{pmatrix}$

• The top number (x) tells you the horizontal movement: positive = right, negative = left
• The bottom number (y) tells you the vertical movement: positive = up, negative = down

Examples:

$\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ means move 3 units right and 4 units up

$\begin{pmatrix} -2 \\ 5 \end{pmatrix}$ means move 2 units left and 5 units up

$\begin{pmatrix} 6 \\ -3 \end{pmatrix}$ means move 6 units right and 3 units down

You can draw a column vector anywhere on a grid, as long as it has the correct length and direction.

Key Rule: Position Vectors Equal Coordinates

If a point has coordinates (a, b), then its position vector is:

$\vec{OA} = \begin{pmatrix} a \\ b \end{pmatrix}$

The position vector has exactly the same numbers as the coordinates.

Example:

• Point A has coordinates (3, 5)
• The position vector is $\vec{OA} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}$

Example:

• The position vector $\vec{OC} = \begin{pmatrix} 7 \\ 2 \end{pmatrix}$
• Therefore, point C has coordinates (7, 2)

Finding a Vector Between Two Points

If you want to find the vector from point A to point B (not starting from the origin), you use this formula:

$\vec{AB} = \vec{OB} - \vec{OA}$

In words: Vector AB = position vector of B minus position vector of A

The formula says: "To get from A to B, go backwards from A to the origin (that's -OA), then go from the origin to B (that's OB)."

Example: Point P has coordinates (2, 3) and point Q has coordinates (5, 7). Find $\vec{PQ}$.

Solution:

• Position vector of P: $\vec{OP} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$
• Position vector of Q: $\vec{OQ} = \begin{pmatrix} 5 \\ 7 \end{pmatrix}$
• $\vec{PQ} = \vec{OQ} - \vec{OP} = \begin{pmatrix} 5 \\ 7 \end{pmatrix} - \begin{pmatrix} 2 \\ 3 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$