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Vector Addition and Geometric Paths

Navigating a city often requires taking multiple connecting streets to reach a destination, whereas a bird can fly there in a single direct line. In geometry, this direct route is the resultant vector, found using vector addition to combine a sequence of known vectors into a single path.

To logically construct an unknown path like $\vec{AC}$ , you must build a "vector journey" through known points, such as $A \to B \to C$ . The resultant vector remains identical regardless of the specific route taken, meaning or .

Worked Example: Constructing a Vector Path

In a triangle $OAB$ , where $\vec{OA} = \mathbf{a}$ and $\vec{OB} = \mathbf{b}$ . Find the vector $\vec{AB}$ in its simplest form.

Step 1: Construct the vector journey. To get from $A$ to $B$ , travel from $A$ to $O$ , then $O$ to $B$ .

\vec{AB} = \vec{AO} + \vec{OB}

Step 2: Substitute the known vectors, remembering to reverse the sign for $\vec{AO}$ . Since $\vec{OA} = \mathbf{a}$ , reversing the direction means .

\vec{AB} = -\mathbf{a} + \mathbf{b}

Step 3: State the final resultant vector.

\vec{AB} = \mathbf{b} - \mathbf{a}

Ratios and Midpoints in Vectors

Dividing a piece of string into a $2:3$ ratio requires cutting it into $5$ equal segments first. In vector geometry, if a point $P$ divides a line segment $AB$ in the ratio $m:n$ , you must divide by the total number of parts () to find the correct fraction of the journey.

The segment $\vec{AP}$ is calculated as $\frac{m}{m+n}\vec{AB}$ . If is the of line , the ratio is exactly , which splits the total journey into two equal parts. This means and .

Worked Example: Finding a Vector via Ratio

Point $P$ lies on the line $AB$ such that $AP:PB = 2:3$ . Given $\vec{OA} = \mathbf{a}$ and , find .

Step 1: Find the full vector $\vec{AB}$ .

\vec{AB} = \vec{AO} + \vec{OB} = -\mathbf{a} + \mathbf{b}

Step 2: Calculate the fractional vector $\vec{AP}$ using the total ratio parts ( $2+3=5$ ).

\vec{AP} = \frac{2}{5}\vec{AB} = \frac{2}{5}(\mathbf{b} - \mathbf{a})

Step 3: Construct the path for $\vec{OP}$ and simplify.

\vec{OP} = \vec{OA} + \vec{AP}

\vec{OP} = \mathbf{a} + \frac{2}{5}(\mathbf{b} - \mathbf{a}) = \frac{3}{5}\mathbf{a} + \frac{2}{5}\mathbf{b}

Scalar Multiples and Parallelism

You can instantly tell two rulers are parallel by looking at them, but proving lines are strictly parallel mathematically requires algebraic conditions. Two vectors are parallel if and only if one is a scalar multiple of the other, meaning one vector can be multiplied by a constant $k$ to equal the other ( $\mathbf{v} = k\mathbf{u}$ ).

The scalar $k$ changes the magnitude (length) of the vector. If $k > 0$ , the vectors point in the same direction, but if $k < 0$ , they point in opposite directions while maintaining parallelism.

To prove parallelism algebraically, factorise both vector expressions to reveal a common vector bracket. For example, $4\mathbf{a} + 6\mathbf{b}$ and $6\mathbf{a} + 9\mathbf{b}$ are parallel because they can be factorised into $2(2\mathbf{a} + 3\mathbf{b})$ and , revealing the shared directional vector .

Worked Example: Proving Parallelism

Given $\vec{MN} = 2\mathbf{a} + 3\mathbf{b}$ and $\vec{XY} = 4\mathbf{a} + 6\mathbf{b}$ , prove that is parallel to .

Step 1: Factorise the larger vector to find a scalar multiple of the smaller vector.

\vec{XY} = 2(2\mathbf{a} + 3\mathbf{b})

Step 2: Substitute the smaller vector into the equation.

\vec{XY} = 2\vec{MN}

Step 3: State the formal conclusion. Since $\vec{XY}$ is a scalar multiple of $\vec{MN}$ , the two vectors are parallel.

Geometric Proofs: Collinearity

If three distant stars perfectly align in the night sky, how do we mathematically prove they lie on the exact same straight line? When three or more points lie on a single straight line, they are collinear.

To construct a causal geometric proof for collinearity, two distinct conditions must be met and explicitly written down. First, you must prove that two vector segments connecting these points are parallel (one is a scalar multiple of the other). Second, you must explicitly state that these two segments share a common point (a shared vertex).

Why are both conditions required? Because if two lines are mathematically parallel and physically pass through the exact same point, they cannot be separate parallel lines — they must be sections of the same straight line.

Worked Example: Proving Collinearity

Given points $O, X,$ and $Y$ where $\vec{OX} = 2\mathbf{a} + \mathbf{b}$ and , prove that and are collinear.

Step 1: Factorise the larger vector to show parallelism.

\vec{OY} = 3(2\mathbf{a} + \mathbf{b})

Step 2: Identify the scalar multiple relationship.

\vec{OY} = 3\vec{OX}

Step 3: Write the final causal proof statement mentioning both conditions. Since $\vec{OY}$ is a scalar multiple of $\vec{OX}$ , the vectors are parallel. Because they both share the common point , the points and are collinear.

Students often mislabel $\vec{AB}$ as $\mathbf{a} - \mathbf{b}$ instead of $\mathbf{b} - \mathbf{a}$ . Always map out the journey $A \to O \to B$ , remembering to reverse the sign of $\vec{OA}$ .

When calculating ratio parts like $1:3$ , students often mistakenly divide by $3$ instead of the total number of parts ( $1 + 3 = 4$ ). Always use as your denominator.

In Edexcel 'explain' questions, always write down your initial "vector journey" equation (e.g., $\vec{MN} = \vec{MO} + \vec{ON}$ ) before substituting any algebra to secure your Method (M) marks.

For collinearity proofs, algebra alone will not get you full marks; you must explicitly write a concluding sentence mentioning both the "scalar multiple" (or parallel nature) and the "common point".

Always expand brackets carefully and simplify your final vector expressions completely (e.g., writing $3\mathbf{a} + \mathbf{b}$ rather than $2\mathbf{a} + \mathbf{a} + \mathbf{b}$ ) to guarantee final accuracy marks.

The single vector representing the sum of two or more vectors; geometrically, it is the direct path from the start of the first vector to the end of the last.

The process of combining a sequence of known vectors to form a continuous vector journey or path.

The point on a line segment that divides it into two equal parts, representing a 1:1 ratio.

A vector multiplied by a real number (scalar), which changes its magnitude and potentially its direction, but keeps it on a parallel line of action.

The geometric condition where two vectors never intersect, proven mathematically when one vector is a scalar multiple of the other.

Three or more points that lie on the exact same straight line.

A shared vertex between two vector segments that structurally connects parallel segments into a single straight line.