Vectors

1. What is a Vector? Notation and Path Rules

Introduction to Vectors

Imagine you leave your house in the morning and walk 3 miles to school. After school, you walk 2 miles to a store to grab a snack, and finally 4 miles all the way back home. In total, your fitness tracker says you’ve walked 9 full miles.

But in terms of vectors, what is your total displacement?

The answer is exactly zero.

Because in vectors, the journey you took doesn’t matter—only where you started and where you finished. Since you ended up right back where you started, your overall change in position is zero. This is the golden rule of vectors: the path doesn’t matter; the only thing that matters is the initial and final position.

Notation & The Golden Path Rule

• Capital Letter Notation: Traveling from point $A$ to $B$ is written as $\vec{AB}$.
• Lowercase Notation: Often given a single bold letter, like $\mathbf{a}$ or $\mathbf{b}$. Handwrite these with an underline!

Whenever you travel against the direction of a given arrow, you must put a minus sign in front of the vector. If you walk backwards along path $B \to A$, the direction has flipped:

2. Adding and Subtracting Vectors

Think of vector addition as taking a journey along designated roads. Sometimes, the road you want isn’t directly available, so you have to take a detour.

Notice how the intermediate letter ($B$) links the two vectors together. Every single vector problem simply comes down to choosing a valid route from your starting point to your finish point and collecting the vectors along the way.

3. Ratios on Line Segments

Handling line segments divided into ratios is one of the most common question formats in higher-tier exams. Here is the step-by-step breakdown:

Worked Example: Finding Vectors with Ratios

Point $P$ lies on the line $AB$ such that $AP:PB = 1:3$. Find $\vec{AP}$ in terms of the full line vector.

SOLUTION

1

Find the full vector first: Ensure you have the full vector of the line containing the ratio, in this case, $\vec{AB}$.

2

Add the ratio parts: The ratio is $1:3$. The total number of parts is $1 + 3 = 4$.

3

Determine the fraction: Since $AP$ is the first part of the ratio, it represents $1$ part out of $4$. Therefore, $\vec{AP} = \frac{1}{4}\vec{AB}$.

4

Build the final route: You can now use $\vec{AP}$ in larger journeys, such as finding $\vec{OP} = \vec{OA} + \vec{AP}$.

4. Geometric Proofs: Collinearity (Straight Lines)

To prove that three points lie on the exact same straight line (collinearity), you must satisfy two distinct conditions:

• Parallel Vectors: Show that two vectors connecting these points are scalar multiples of each other.
• Shared Point: State that the two vectors share a common point.

If two lines are parallel and they share a point, they cannot be two separate lines. They must be one single straight line!

The Extended Triangle Proof

In triangle $OAB$, $\vec{OA} = 5\mathbf{a}$ and $\vec{OB} = 3\mathbf{b}$.
• Point $C$ is on $OA$ such that $OC:CA = 4:1$.
• $M$ is the midpoint of $AB$.
• $D$ is a point on extended line $OB$ such that $OB:OD = 3:4$.
Show that points $C$, $M$, and $D$ lie on the same straight line.

SOLUTION

1

Establish position vectors:
Line $OA$ ratio is $4:1$ (total 5). $$ \vec{OC} = \frac{4}{5}(5\mathbf{a}) = 4\mathbf{a} $$ Line $OBD$ ratio is $3:4$. Since $\vec{OB} = 3\mathbf{b}$, each part is $\mathbf{b}$, so: $$ \vec{OD} = 4\mathbf{b} $$

2

Find the midpoint vector $\vec{AM}$:
First, find the full line vector $\vec{AB}$: $$ \vec{AB} = \vec{AO} + \vec{OB} = -5\mathbf{a} + 3\mathbf{b} $$ Since $M$ is the midpoint, we halve this vector: $$ \vec{AM} = \frac{1}{2}(-5\mathbf{a} + 3\mathbf{b}) = -2.5\mathbf{a} + 1.5\mathbf{b} $$

3

Calculate First Test Vector ($\vec{CM}$):
Determine the route: $$ \vec{CM} = \vec{CA} + \vec{AM} $$ $\vec{CA}$ is the remaining $1$ part of $OA$, so $\vec{CA} = \mathbf{a}$. Substituting this in: $$ \vec{CM} = \mathbf{a} + (-2.5\mathbf{a} + 1.5\mathbf{b}) = -1.5\mathbf{a} + 1.5\mathbf{b} $$ Factor out $1.5$ to simplify: $$ \vec{CM} = 1.5(-\mathbf{a} + \mathbf{b}) $$

4

Calculate Second Test Vector ($\vec{CD}$):
Determine the route: $$ \vec{CD} = \vec{CO} + \vec{OD} $$ Substitute the known vectors: $$ \vec{CD} = -4\mathbf{a} + 4\mathbf{b} $$ Factor out $4$ to simplify: $$ \vec{CD} = 4(-\mathbf{a} + \mathbf{b}) $$

5

Final Conclusion:
Comparing our two test vectors: $$ \vec{CM} = 1.5(-\mathbf{a} + \mathbf{b}) $$ $$ \vec{CD} = 4(-\mathbf{a} + \mathbf{b}) $$ We can clearly see they are scalar multiples of the exact same bracket. Therefore, the vectors are parallel. Because they also share the common point $C$, the points $C$, $M$, and $D$ must lie on the same straight line.