Vectors Proof: Parallel Vectors & Collinear Points

What are vector proofs?

Vector proofs use vector expressions and properties to prove geometrical facts like:

• Whether vectors are parallel
• Whether points lie on a straight line (collinear)
• How to divide a vector in a given ratio

Parallel Vectors

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

$\mathbf{b} = k\mathbf{a} \quad \text{for some scalar } k$

• If $k > 0$, they're in the same direction
• If $k < 0$, they're in opposite directions

Example:

$$\mathbf{a} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 2 \\ 6 \end{pmatrix} = 2\mathbf{a}$$

So $\mathbf{a}$ and $\mathbf{b}$ are parallel.

If the two vectors factorise and have a bracket that is common then they are also parallel. For example

• $6\mathbf{b} - 10\mathbf{a} \to 2(3\mathbf{b} - 5\mathbf{a})$
• $12\mathbf{b} - 20\mathbf{a} \to 4(3\mathbf{b} - 5\mathbf{a})$
• So $12\mathbf{b} - 20\mathbf{a} = 2(6\mathbf{b} - 10\mathbf{a})$

Collinear Points

To show three points lie on a straight line (are collinear):

Show two of the vectors (e.g., $\vec{AB}$ and $\vec{BC}$) are parallel

Make sure the segments are connected (they share a point)

If: $\vec{BC} = 2\vec{AB}$ Then $A$, $B$, and $C$ lie on the same line.

Parallel isn’t enough — you need shared points too

Using Ratios with Vectors

If a line is divided in a ratio (e.g. $AP : PC = 3:1$), the total parts = $3 + 1 = 4$

• $\vec{AP} = \frac{3}{4}\vec{AC}$
• $\vec{PC} = \frac{1}{4}\vec{AC}$

This helps us express positions and midpoints accurately.

Example

Trapezium OABC

Given:

• $\vec{OA} = 2\mathbf{a}$
• $\vec{OC} = \mathbf{c}$
• $\vec{AB} = 3\vec{OC} = 3\mathbf{c}$

and AB is parallel to OC,

(a) Find $\vec{OB}$ and $\vec{AC}$

$$\vec{OB} = \vec{OA} + \vec{AB} = 2\mathbf{a} + 3\mathbf{c} \vec{AC} \\ = -\vec{OA} + \vec{OC} = -2\mathbf{a} + \mathbf{c} = \mathbf{c} - 2\mathbf{a}$$

(b) Point P lies on AC such that $AP : PC = 3 : 1$

Total parts = 4

$$\vec{AP} = \frac{3}{4}\vec{AC} = \frac{3}{4}(\mathbf{c} - 2\mathbf{a}) = -\frac{3}{2}\mathbf{a} + \frac{3}{4}\mathbf{c}$$

$$\vec{OP} = \vec{OA} + \vec{AP} = 2\mathbf{a} + \left( -\frac{3}{2}\mathbf{a} + \frac{3}{4}\mathbf{c} \right) = \frac{1}{2}\mathbf{a} + \frac{3}{4}\mathbf{c}$$

(c) Show that P lies on OB and find the ratio $OP : PB$

From earlier: $\vec{OB} = 2\mathbf{a} + 3\mathbf{c}$

Compare: $\vec{OP} = \frac{1}{4}(2\mathbf{a} + 3\mathbf{c}) = \frac{1}{4}\vec{OB}$

So OP is $\frac{1}{4}$ of OB. That means: $OP : PB = 1 : 3$

Conclusion:

• $\vec{OP}$ and $\vec{OB}$ are parallel
• They share point O
• So P is on the line OB
• The ratio is $1:3$