Perpendicular lines — product of slopes = −1: Foundations

This page assumes you have seen nothing. Every letter, ratio, and picture the parent note uses is built here from the ground up, in the order that lets each piece rest on the one before it.

1. The plane, and what a point $(x, y)$ means

Before any line, we need a place to draw. Imagine two number lines crossing at right angles: one running left–right (the == $x$ -axis) and one running up–down (the $y$ -axis). Where they cross is the origin==, written $O = (0, 0)$ .

A point is a single dot in this plane. We name it with two numbers in a bracket, $(x, y)$ :

the first number $x$ says how far right (positive) or left (negative) of the origin,
the second number $y$ says how far up (positive) or down (negative).

Why the topic needs this: every line, slope, and angle lives on this grid. Points like $A = (1, m_{1})$ in the parent's Pythagoras proof are just addresses on this plane.

2. Rise and run — the two moves that make a slope

Pick any two points on a line. To get from one to the other you make two moves:

run = the horizontal move = change in $x$ ,
rise = the vertical move = change in $y$ .

The little triangle symbol $Δ$ (Greek capital "delta") just means "change in". So: $Δ x = x_{2} - x_{1} (run), Δ y = y_{2} - y_{1} (rise) .$

Why the topic needs this: slope is built as $\frac{rise}{run} = \frac{Δ y}{Δ x}$ . Without rise and run there is no slope.

3. Slope $m$ — one number for steepness

$m = 2$ : walk $1$ right, go $2$ up — steep, climbing.
$m = \frac{1}{2}$ : walk $1$ right, go half a step up — gentle climb.
$m = 0$ : walk right, go nowhere up — flat / horizontal.
$m = - 3$ : walk $1$ right, go $3$ down — descending.

The vertical case (a warning that returns later): a straight up-and-down line has run $Δ x = 0$ . Then $m = \frac{Δ y}{0}$ , and dividing by zero is undefined. This is why the parent note treats vertical lines as a special case — a vertical line simply has no slope number.

Recall Check: what is the slope through

$(1, 2)$ and $(4, 3)$ ? $Δ y = 3 - 2 = 1$ , $Δ x = 4 - 1 = 3$ , so $m = \frac{1}{3}$ . This is exactly Example 4 in the parent note.

See Slope of a Line for the full treatment.

4. The inclination angle $θ$ and why $m = tan θ$

Instead of "rise over run," we can describe a line's tilt by the angle it makes with the positive $x$ -axis, swept anticlockwise. This angle is written $θ$ (Greek "theta").

Drop the rise and run of the line and you get a right triangle: run along the bottom, rise going up, the line itself as the slanted side (the hypotenuse). Inside this triangle:

the side across from $θ$ (the rise) is called the opposite,
the side next to $θ$ (the run) is called the adjacent.

Why this tool and not another? We want to trade a ratio (slope) for an angle (tilt), because "perpendicular" is naturally an angle idea ( $90°$ ). The tangent is the exact function that converts an angle into the rise/run ratio — sine or cosine alone would give you only one side, not the ratio we need. This bridge is what powers Way 2 of the parent's proof, where $tan (θ + 90°)$ appears.

5. Perpendicular and the right angle $90°$

Two lines are perpendicular when they cross making a right angle: a perfect square corner, written $90°$ (or the small square symbol $└$ ).

Why the topic needs this: the entire note is a test for this relationship — given slopes, decide if the angle between the lines is $90°$ .

See Angle Between Two Lines; perpendicular is the special case where that angle equals $90°$ .

6. Direction vectors and rotation by $90°$

An arrow from one point to another, capturing which way and how far, is a direction vector, written like $(Δ x, Δ y)$ — run first, rise second. A line pointing "1 right, 2 up" has direction vector $(1, 2)$ , and slope $\frac{Δ y}{Δ x} = \frac{2}{1} = 2$ .

To make a line perpendicular, we rotate its direction vector by $90°$ anticlockwise. The rule the parent note uses is: $(x, y) \mapsto (- y, x) .$

Here $\mapsto$ ("maps to") just means "becomes." Why this exact rule? Test it on the two axis arrows:

$(1, 0)$ (pointing right) $\mapsto (0, 1)$ (now pointing up) ✔ a quarter-turn.
$(0, 1)$ (pointing up) $\mapsto (- 1, 0)$ (now pointing left) ✔ a quarter-turn.

If a quarter-turn does the right thing to both building-block directions, it does the right thing to every direction. Details live in Rotation of Vectors by 90°.

Why the topic needs this: applying $(x, y) \mapsto (- y, x)$ to $(Δ x, Δ y)$ gives $(- Δ y, Δ x)$ , whose slope is $- \frac{Δ x}{Δ y}$ — the negative reciprocal. That is the shortest road to $m_{1} m_{2} = - 1$ .

7. Pythagoras and distance — the trig-free road

For a right triangle with legs $a$ , $b$ and hypotenuse $c$ , Pythagoras Theorem says $a^{2} + b^{2} = c^{2} .$

To measure the length between two points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ , we treat the horizontal and vertical gaps as the two legs and the straight-line distance as the hypotenuse — this gives the Distance Formula: $distance = (x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} .$

Why the topic needs this: Way 3 of the parent proof places points $A = (1, m_{1})$ and $B = (1, m_{2})$ and asks when is the corner at $O$ a right angle? — answered by checking $∣ O A ∣^{2} + ∣ O B ∣^{2} = ∣ A B ∣^{2}$ . See Pythagoras Theorem and Distance Formula.

8. Equation of a line and the point–slope form

A line can be written as $y = m x + c$ , where $m$ is the slope (coefficient of $x$ ) and $c$ is where the line crosses the $y$ -axis. When you know one point $(x_{0}, y_{0})$ and the slope $m$ , the fastest form is point–slope: $y - y_{0} = m (x - x_{0}) .$

Why the topic needs this: Example 3 of the parent builds a perpendicular line through a given point using exactly this form. See Equation of a Line — point-slope form.

(x, y)

tell you to do from the origin?

Walk

x

steps horizontally, then

y

steps vertically.

What does the symbol

Δ

mean?

"Change in" — new value minus old value.

Define slope in words and as a formula.

Rise over run;

m = \frac{Δ y}{Δ x}

Why is a vertical line's slope undefined?

Its run

Δ x = 0

, and dividing by zero has no value.

On the inclination triangle,

tan θ = ?

opposite over adjacent = rise over run = the slope.

State the bridge between slope and angle.

m = tan θ

What is a right angle in degrees, and as a fraction of a full turn?

90°

; one quarter of a full turn.

What is the

90°

anticlockwise rotation rule for a vector?

(x, y) \mapsto (- y, x)

State Pythagoras' Theorem.

a^{2} + b^{2} = c^{2}

for legs

a, b

and hypotenuse

c

Write the distance between

(x_{1}, y_{1})

and

(x_{2}, y_{2})

(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Write the point–slope form of a line.

y - y_{0} = m (x - x_{0})

What relates the slopes of parallel lines?

They are equal:

m_{1} = m_{2}

Connections

Slope of a Line — the definition $m = tan θ$ built here.
Parallel Lines — equal slopes — the twin rule.
Equation of a Line — point-slope form — used to build perpendicular lines.
Angle Between Two Lines — perpendicular is its $90°$ case.
Rotation of Vectors by 90° — powers the cleanest proof.
Distance Formula & Pythagoras Theorem — power the trig-free proof.
Parent: Perpendicular Lines

Foundations — Perpendicular lines — product of slopes = −1