10.4 Position vectors and the distance between two points — Learn

Prerequisites: plotting coordinates in two and three dimensions; subtracting directed numbers; finding the magnitude of a vector (Section 10.2); adding vectors (Section 10.3).

By the end of this section you can: write the position vector of a point; find the displacement between two points from their position vectors; find the distance between two points in two and three dimensions; find the midpoint of two points.

1. Position vectors

Choose an origin $O$. The position vector of a point $A$ is the vector from $O$ to $A$. It is written $\vec{OA}$ or $\mathbf{a}$.

The position vector of $A(x, y)$ is

In three dimensions, the position vector of $A(x, y, z)$ has a third component. The position vector tells you where the point is, so it depends on the origin. A free vector does not.

Example 1. Write the position vectors of $A(1, 2)$ and $B(5, 7)$.

2. The displacement between two points

To travel from $A$ to $B$, first go from $A$ back to the origin, then from the origin out to $B$. In vectors,

So the displacement from $A$ to $B$ is the position vector of $B$ minus the position vector of $A$.

Example 2. Find $\vec{AB}$ for $A(1, 2)$ and $B(5, 7)$.

Example 3. For $A(1, 2, -1)$ and $B(4, -2, 3)$, find $\vec{AB}$.

3. The distance between two points

The distance between $A$ and $B$ is the magnitude of $\vec{AB}$:

in two dimensions, with a third square for the $z$-coordinates in three dimensions. This is the same formula as the distance between two points from coordinate geometry, now written with vectors.

Example 4. Find the distance between $A(1, 2)$ and $B(5, 7)$.

From Example 2, $\vec{AB} = 4\mathbf{i} + 5\mathbf{j}$, so

Example 5. Find the distance between $A(1, 2, -1)$ and $B(4, -2, 3)$.

From Example 3, $\vec{AB} = 3\mathbf{i} - 4\mathbf{j} + 4\mathbf{k}$, so

The two distances are equal here, which is a coincidence of these coordinates.

Example 6. Find the distance of $P(3, 4, 12)$ from the origin.

The position vector of $P$ is $3\mathbf{i} + 4\mathbf{j} + 12\mathbf{k}$, so

4. The midpoint of two points

The midpoint $M$ of $A$ and $B$ has position vector

This is the average of the two position vectors, taken component by component.

Example 7. Find the midpoint of $A(1, 2)$ and $B(5, 7)$.

The midpoint is $\left(3, \frac{9}{2}\right)$.

Watch out. Three slips are common on this page.

The subtraction in the wrong order. $\vec{AB} = \mathbf{b} - \mathbf{a}$, not $\mathbf{a} - \mathbf{b}$. The wrong order gives $\vec{BA}$, which points the opposite way. The distance is unaffected, because squaring removes the signs, but the displacement vector is wrong.

A vector answer to a distance question. The distance is a single number, not a column. Read the question: "find $\vec{AB}$" wants a vector, and "find the distance" wants a magnitude.

The origin left out. A position vector is measured from a stated origin. If the origin changes, the position vectors change, even though the distance between the two points does not. Re-check: for a distance, subtract the coordinates first, then square. Do not add the two position vectors by mistake.

Progress check

Return to the chapter problem. The drone finishes at $10\mathbf{i} + 3\mathbf{j}$ and the walker is at $\mathbf{w} = 13\mathbf{i} + 7\mathbf{j}$.

1. Find the displacement from the drone to the walker.
2. Find the distance between the drone and the walker.
3. Find the midpoint of the drone and the walker.

You should now be able to

• write the position vector of a point in two or three dimensions;
• find the displacement between two points from their position vectors;
• find the distance between two points, using the correct formula;
• find the midpoint of two points from their position vectors.