Prove and use coordinate line criteria

Try

A line passes through P=(2,−1) with direction (3,4). Moving three units horizontally and four vertically reaches Q=(5,3), so its slope is 4/3. A perpendicular direction is (−4,3): a quarter-turn rotates the original displacement without changing its length.

The perpendicular through P therefore has slope −3/4. These fractions are negative reciprocals, but the rule needs an explanation and exceptions. A vertical line has no finite slope, and a horizontal line is perpendicular to it even though multiplying two ordinary slope numbers is impossible.

Predict

For a general nonzero direction (u,v), a 90-degree counterclockwise rotation gives (−v,u). Predict its slope when u and v are both nonzero: the original is v/u and the rotated slope is −u/v, whose product is −1.

If v=0, the original line is horizontal and the rotated direction is vertical. If u=0, the original is vertical and its perpendicular is horizontal. Handling these cases directly is more informative than trying to assign infinity an ordinary numerical slope.

Experiment

Construct both worked lines using two points each. The original line contains P=(2,−1) and Q=(5,3). The perpendicular contains P and R=(−2,2). Draw them on equal-scale axes; a stretched graph can make a right angle look wrong.

Line	Direction	Equation
Original through P	(3,4)	y+1=(4/3)(x−2)
Perpendicular through P	(−4,3)	y+1=−(3/4)(x−2)
Parallel through (0,2)	(3,4)	y−2=(4/3)x

Verify the listed points in their equations. Next construct a vertical line through (−1,2) and its perpendicular through (3,−4): the equations are x=−1 and y=−4. The two lines need not meet at either supplied point.

Understand

A line through P can be described by P+t(u,v), where t ranges over all real numbers and (u,v) is nonzero. Two direction vectors are scalar multiples precisely when their lines have the same direction. If the lines are distinct, this is parallelism; if they share a point, they coincide. Equal slopes alone do not distinguish those cases.

For nonvertical lines, equality of slopes v/u=s/r gives vr=su. This makes (r,s) a nonzero multiple of (u,v), so the lines have the same direction. Conversely proportional direction vectors give equal slopes. Two vertical lines also share a direction; they are distinct and parallel exactly when their fixed x-coordinates differ.

For perpendicularity, rotate a direction by a quarter-turn. Rigid rotation gives a right angle by construction. If both slopes exist and are nonzero, their product is −1 as calculated above. Conversely, suppose two finite slopes m and n satisfy mn=−1. The first line has direction (1,m). Its quarter-turn direction is (−m,1), with slope −1/m=n, so the second line is parallel to this perpendicular direction and is itself perpendicular to the first. This proves both directions rather than treating the product rule as a definition.

The coordinate argument uses established rigid quarter-turns and uniqueness of a line direction. It does not require a dot-product course. It also covers a line with a negative slope: the sign and order of the rotated components matter.

To construct a parallel or perpendicular through a specified point, retain the appropriate direction and change only the anchor point. In slope form use y−y₀=m(x−x₀), but use x=x₀ for a vertical result. Substitution then checks that the chosen point actually lies on the constructed line.

Apply

Solve a fresh through-point construction with a non-axis-aligned supplied line, and another with a vertical or horizontal supplied line. Provide the new equation, two constructed points and a reason that the direction has the requested relationship.

An invalid proof says two lines are perpendicular because their slopes are reciprocals. The negative sign is essential when both slopes are finite and nonzero. Another invalid proof calls y=2x+1 and 2y=4x+2 parallel distinct lines; they are the same line. Always compare both direction and position.

Continue

Direction reasoning supports coordinate proofs of rectangles, heights and polygon areas. It also underlies the parallel auxiliary lines used to divide a segment in a specified ratio.

Optional depth: a line can be written ax+by=c with (a,b) nonzero. A direction is (b,−a), which keeps ax+by unchanged. This representation includes vertical lines without a special undefined slope and gives another way to check parallel and perpendicular directions.