Vector Proof (Edexcel IGCSE Maths B): Revision Note

Vector proof

What are vector proofs?

• Vectors can be used to prove things that are true in geometrical diagrams

  • Vector proofs can be used to find additional information that can help us to solve problems

How do I know if two vectors are parallel?

• Two vectors are parallel if one is a scalar multiple of the other

  • This means if b is parallel to a, then b = ka

    •  where k is a constant number (scalar)

• For example, $\mathbf{a}=\left(\begin{array}{c}1\\ 3\end{array}\right)$ and $\mathbf{b}=\left(\begin{array}{c}2\\ 6\end{array}\right)$ 

  • $\left(\begin{array}{c}2\\ 6\end{array}\right)=2\times \left(\begin{array}{c}1\\ 3\end{array}\right)$ so $\mathbf{b}=2\mathbf{a}$

  • b is a scalar multiple of a, so b is parallel to a

• If the scalar multiple is negative, then the vectors are parallel and in opposite directions

  • $\mathbf{c}=\left(\begin{array}{c}-3\\ -9\end{array}\right)=-3\mathbf{a}$

    • c is parallel to a and in the opposite direction

• If two vectors factorise with a common bracket, then they are parallel

  • They can be written as scalar multiples 

• For example

  • 9a + 6b factorises to 3(3a + 2b)

  • 12a + 8b factorises to 4(3a + 2b)

  • This means $12\mathbf{a}+8\mathbf{b}=\frac{4}{3}\left(9\mathbf{a}+6\mathbf{b}\right)$

    • so they are scalar multiples of each other

    • and therefore parallel

How do I know if three points lie on a straight line?

• You may be asked to prove that three points lie on a straight line

  • Points that lie on a straight line are collinear

• To show that the points A , B  and C  are collinear

  • prove that two line segments are parallel

  • and show that there is at least one point that lies on both segments

    • This makes them parallel and connected (not parallel and side-by-side)

• For example, if you show that $\overrightarrow{BC}=2\overrightarrow{AB}$ then

  • the line segments AB  and BC  are parallel

  • and they have a common point, B 

    • So A , B  and C  must be collinear

• Similarly, $\overrightarrow{AC}=3\overrightarrow{AB}$ means AC  and AB  are parallel

  • and they have a common point, A

    • so A , B  and C  must be collinear

How do I use ratios in vector paths?

• Convert ratios into fractions

• In the example shown, if $AX:XB=3:5$ then

  • $\overrightarrow{AX}=\frac{3}{8}\overrightarrow{AB}$

  • $\overrightarrow{\mathit{X}\mathit{B}}=\frac{5}{8}\overrightarrow{AB}$

    • The ratio 3:5 has 3 + 5 = 8 parts

• Always check which ratio you are being asked for

  • $\overrightarrow{\mathit{A}\mathit{X}}=\frac{3}{5}\overrightarrow{XB}$

  • $\overrightarrow{XB}=\frac{5}{3}\overrightarrow{AX}$