Parallel and perpendicular lines

Two straight lines in the same coordinate system either cross each other or run alongside each other without ever meeting. If we write both lines as $y = mx + c$, the gradients tell us which of the two it is, and whether the crossing happens at a right angle.

When do I use this?

When a question gives you the equations of two lines and asks whether they're parallel or perpendicular, or when you have to write the equation of a line that must be parallel or perpendicular to another one. Everything here rests on what the gradient means, which you can read about in gradient and y-intercept, and it belongs to coordinate geometry.

Parallel lines: the same gradient

The gradient of a linear function shows how much we go up the y-axis when we go $1$ along the x-axis. So if two lines go in the same direction, they must also have the same gradient. That's what parallel lines are: two lines with the same gradient. Take

$$y = 3x + 6 \quad \text{and} \quad y = 3x + 1$$

Both go up $3$ for every $1$ along, so they run alongside each other and never cross. The only difference between them is where they cross the y-axis.

If the gradients are different, the lines cross exactly once, and we can find where. Say we have $y = 3x + 6$ and $y = 2x + 4$. We set the two functions equal to each other, which we're only allowed to do because both of them equal $y$:

$$3x + 6 = 2x + 4$$

Solving this gives $x = -2$. That's the x-coordinate of the crossing point. To find the y-coordinate we put $x = -2$ into one of the functions, and it doesn't matter which one:

$$y = 3 \times (-2) + 6 = 0$$

So the two lines cross at $(-2, 0)$. If we draw them in a coordinate system, we can see that it fits. Two parallel lines never give us a crossing point like this.

Perpendicular lines: gradients that multiply to -1

First, what are perpendicular lines? In my Danish notes they're called orthogonal lines, and orthogonal is just a smart word for right-angled. It means the two lines stand at right angles to each other, so they make an angle of $90°$. There's a rule, a kind of formula, that tells us something about two lines when they're perpendicular. It says that when two lines are perpendicular, the product of their gradients is $-1$:

$$m_1 \times m_2 = -1$$

Here $m_1$ is the gradient of one line and $m_2$ is the gradient of the other. The rule really just says: if you have two lines and multiply their gradients together, you can see whether they're perpendicular. If it gives $-1$, they are.

Let's take an example. We're given two lines,

$$y = 3x + 5 \quad \text{and} \quad y = -4x - 2$$

and want to find out whether they're perpendicular. We use the rule, put in the two gradients and work it out:

$$3 \times (-4) = -12$$

Since it gives $-12$ and not $-1$, the lines are not perpendicular. The two lines on the figure, on the other hand, have gradients $2$ and $-0.5$, and $2 \times (-0.5) = -1$, so those two really are perpendicular.

Using the rule as a condition

We often use the rule as a kind of condition when we solve problems. It could be a task where we have to find the equation of a line that we know must be perpendicular to another line. Then we also know that the two gradients have to satisfy the rule: their product must be $-1$.

Say a line has gradient $3$, and we want a line perpendicular to it. Its gradient $m_2$ has to satisfy

$$3 \times m_2 = -1$$

so $m_2 = -\tfrac{1}{3}$. Any line with gradient $-\tfrac{1}{3}$ is perpendicular to our line. If we're also given a point the new line has to pass through, we can write its equation with $y = y_0 + m(x - x_0)$ from the coordinate geometry glossary.

Where does the rule come from? In the book it's derived with vectors, which is beyond GCSE, so here we just use it.

Common mistakes

• "The gradients only need opposite signs." No. $3$ and $-4$ have opposite signs, but $3 \times (-4) = -12$, so those lines aren't perpendicular. The product has to be exactly $-1$.
• "Parallel lines have the same $c$." It's the gradient $m$ they share. $y = 3x + 6$ and $y = 3x + 1$ are parallel and cross the y-axis in different places.
• Setting two lines equal without checking they're both $y =$. We're only allowed to write $3x + 6 = 2x + 4$ because both sides equal $y$. If one of the lines is written differently, get it into that form first.

Related

• Coordinate geometry: lines, distances and circles
• Gradient and y-intercept: what m and c mean
• Finding the equation of a straight line from two points