Normal vector and orthogonality

Proving that an angle is a right angle

The dot product is the most effective tool for proving that a triangle is a right-angled triangle or that two lines are perpendicular: one simply needs to show that the dot product of the direction vectors is 0.

Example: Let A(1, 2), B(4, 3) and C(2, -1). Is triangle ABC a right-angled triangle at A? AB(3; 1) and AC(1; -3). AB · AC = 3 × 1 + 1 × (-3) = 3 − 3 = 0. Therefore, AB and AC are perpendicular: the triangle is a right-angled triangle at A.

The normal vector to a line

A vector n(a; b) is normal to a line d if it is orthogonal to every direction vector of d. This concept provides a very quick method for finding the equation of a line.

If n(a; b) is normal to d and if A(x₀; y₀) lies on d, then for any point M(x; y) on d:

AM·n = 0, that is, a*(x – x₀) + b*(y – y₀) = 0

This gives the Cartesian equation of d in the form ax + by + c = 0, where the coefficients a and b are directly given by the coordinates of a normal vector.

Example of a line equation

We want the equation of the line passing through A(2; -1) and with normal vector n(3; 4). 3*(x - 2) + 4*(y + 1) = 0, or 3x + 4y - 2 = 0.

Common pitfall

Do not confuse the normal vector with the direction vector! If n(a; b) is normal to d, then a direction vector of d is u(-b; a) (we swap the coordinates and change one sign). Mixing the two up means you lose all the marks.