Rukasha Maths

GCSE maths Beta

Vectors and geometric proof · Revision guide

Higher · Geometry · 2 pages in the PDF

Download PDF2 pagesRead the study notes

Learning objectives

  • Add, subtract and reverse vectors along a path.
  • Prove parallelism using a non-zero scalar multiple.

Key ideas

Subtract position vectors to find a displacement.

OA→=a,OB→=b⇒AB→=b−a\overrightarrow{OA}=\mathbf a,\quad\overrightarrow{OB}=\mathbf b\quad\Rightarrow\quad\overrightarrow{AB}=\mathbf b-\mathbf a

Following a path adds vectors; reversing direction changes the sign.

BA→=−AB→\overrightarrow{BA}=-\overrightarrow{AB}

A midpoint averages the two position vectors.

OM→=a+b2\overrightarrow{OM}=\frac{\mathbf a+\mathbf b}{2}

For non-zero vectors, a non-zero scalar multiple proves parallelism. A positive scalar preserves direction.

Worked examples

Example 1

O and A are distinct. M is the midpoint of AB; N is the midpoint of OB. Prove NM is parallel to OA and half its length.

OA→=a,OB→=b\overrightarrow{OA}=\mathbf a,\quad\overrightarrow{OB}=\mathbf b
OM→=a+b2,ON→=b2\overrightarrow{OM}=\frac{\mathbf a+\mathbf b}{2},\qquad\overrightarrow{ON}=\frac{\mathbf b}{2}
NM→=OM→−ON→\overrightarrow{NM}=\overrightarrow{OM}-\overrightarrow{ON}
NM→=a2=12OA→\overrightarrow{NM}=\frac{\mathbf a}{2}=\frac12\overrightarrow{OA}

Example 2

M is the midpoint of AB. Find the vector from A to M.

AB→=b−a\overrightarrow{AB}=\mathbf b-\mathbf a
AM→=b−a2\overrightarrow{AM}=\frac{\mathbf b-\mathbf a}{2}

Watch out

Keep the order of subtraction correct. Position vectors start at the origin; a displacement vector may start elsewhere.