Vector geometry
Vector geometry (Extended)
- The position vector of a point is the vector from the origin $O$ to it.
- The vector from $A$ to $B$ is
$$\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$$
- Two vectors are parallel if one is a scalar multiple of the other. Points are collinear if the vectors between them are parallel and share a point.
Practice
If a and b are the position vectors of A and B, then the vector AB is:
AB = (position of B) − (position of A) = b − a.
Practice
Two vectors are parallel if one is a scalar multiple of the other.
Parallel vectors point the same (or opposite) way, so one is k times the other.
Worked example: a midpoint
- $\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OB}=\mathbf{b}$; $M$ is the midpoint of $AB$.
- $\overrightarrow{OM} = \mathbf{a} + \tfrac12\overrightarrow{AB} = \mathbf{a} + \tfrac12(\mathbf{b}-\mathbf{a}) = \tfrac12(\mathbf{a}+\mathbf{b})$.
Practice
M is the midpoint of AB. The position vector OM is:
OM = a + ½(b − a) = ½(a + b).
You've got it
Key idea
- $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$ (end minus start)
- parallel vectors are scalar multiples; that's how you show points are collinear
- midpoint of $AB$: $\overrightarrow{OM} = \tfrac12(\mathbf{a}+\mathbf{b})$