Identifying orthogonal vectors

A very powerful criterion

We have seen that if u and v form an angle of 90 degrees, then cos(90) = 0, so u·v = 0. This result also holds in the opposite direction: this is known as the orthogonality criterion.

u and v are orthogonal <=> u·v = 0

This criterion is very useful because it allows us to check whether two vectors are perpendicular with a simple calculation, without needing to measure an angle with a protractor.

Example using coordinates

Let u(3; -2) and v(4; 6). We calculate:

u·v = 3·4 + (-2)·6 = 12 - 12 = 0

As u·v = 0, the vectors u and v are orthogonal: if we plot them, the lines they lie on are perpendicular.

Application: proving that a triangle is right-angled

To show that a triangle ABC is right-angled at A, it suffices to calculate the coordinates of the vectors AB and AC, then check that AB·AC = 0. If the dot product is zero, the triangle is right-angled at A; otherwise, it is not.

Application: Equation of a line using a normal vector

A normal vector n(a; b) to a line d is a vector orthogonal to all the direction vectors of d. A point M(x; y) lies on d (which passes through a point A) if and only if the vector AM is orthogonal to n, that is, AM·n = 0. This gives an equation of the form:

a*(x - x_A) + b*(y - y_A) = 0, or ax + by + c = 0

Common pitfall

Note that u·v = 0 means that u and v are orthogonal ONLY if both u and v are non-zero. If one of the two vectors is the zero vector, the dot product is also 0, but in that case nothing can be said about an angle (the zero vector has no direction).