13.2 Position and Unit Vectors

📖 Main Content

Part 1: Position Vectors

What is a Position Vector?

A position vector tells you where a point is located relative to a fixed starting point called the origin, O.

Think of it like giving directions from the centre of a town (the origin) to your house (the point). The position vector is the "journey" from O to that point.

• The position vector of point $P$ from origin $O$ is written as $\vec{OP}$, and is often just called p (a bold lowercase letter).
• The position vector of point $A$ from origin $O$ is written as $\vec{OA}$, or simply a.
• The position vector of point $B$ from origin $O$ is written as $\vec{OB}$, or simply b.

Finding the Vector Between Two Points

Now imagine you know where point $A$ is (its position vector a) and where point $B$ is (its position vector b). How do you find the vector that goes directly from A to B?

You use this key formula:

$\vec{AB} = \vec{OB} - \vec{OA}$

Or using bold letter shorthand:

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

Why does this work? Think of it as a two-step journey:

• First, go backwards from $A$ to the origin $O$ — that is $-\mathbf{a}$
• Then, go forwards from $O$ to $B$ — that is $+\mathbf{b}$
• Combined: $\vec{AB} = \mathbf{b} - \mathbf{a}$ ✓

Finding the Position Vector of a Point on a Line Segment

Sometimes a question tells you that a point $R$ lies somewhere between two known points $P$ and $Q$ on a line. You need to find where $R$ is (its position vector from $O$).

The method: Use the chain rule for vectors:

$\vec{OR} = \vec{OP} + \vec{PR}$

This means: start at O, go to P, then go from P to R.

📝 Worked Example 1

Part a — Finding $\vec{PQ}$:

Use the formula $\vec{PQ} = \vec{OQ} - \vec{OP}$:

$\vec{PQ} = (10\mathbf{i} - 3\mathbf{j}) - (4\mathbf{i} + 5\mathbf{j})$

Subtract the $\mathbf{i}$ components: $10 - 4 = 6$

Subtract the $\mathbf{j}$ components: $-3 - 5 = -8$

$\boxed{\vec{PQ} = 6\mathbf{i} - 8\mathbf{j}}$

Part b — Finding the position vector of $R$:

Step 1: Find $\vec{PR}$ — we're told $R$ is $\frac{1}{4}$ of the way from $P$ to $Q$:

$\vec{PR} = \frac{1}{4}\vec{PQ} = \frac{1}{4}(6\mathbf{i} - 8\mathbf{j}) = 1.5\mathbf{i} - 2\mathbf{j}$

Step 2: Find the position vector of $R$ from $O$:

$\vec{OR} = \vec{OP} + \vec{PR} = (4\mathbf{i} + 5\mathbf{j}) + (1.5\mathbf{i} - 2\mathbf{j})$

$\boxed{\vec{OR} = 5.5\mathbf{i} + 3\mathbf{j}}$