Perpendicular = dot product zero
Orthogonality is the grown-up word for perpendicular, and its test is the cleanest in all of linear algebra: two vectors are orthogonal exactly when their dot product is zero. No angles to measure, no pictures to draw — just one arithmetic check.
(2,1) dot (-1,2) = 2*(-1) + 1*2 = -2 + 2 = 0 --> these two vectors are orthogonal
Why right angles carry independent information
When directions are orthogonal, none of them leaks into the others. Move along the east-west axis and your north-south coordinate does not budge. That is why a map uses perpendicular axes: each one reports something the other cannot. Tilted, non-perpendicular axes still work, but readings on one contaminate the other.
The orthonormal gold standard
Take orthogonal directions and shrink each to length 1. Now they are not only perpendicular but unit length, and we call the collection an orthonormal basis — 'ortho' for the right angles, 'normal' for the unit lengths. It is the basis everyone wishes they had, because computing coordinates becomes trivial: a coordinate is just a dot product.
e1 = (1,0) e2 = (0,1) e1 dot e2 = 0 (orthogonal) norm(e1) = 1, norm(e2) = 1 (unit length) --> e1, e2 form an orthonormal basis
Stack an orthonormal basis as the columns of a matrix and you get an orthogonal matrix Q. Such a matrix is a perfect rigid motion: it rotates or reflects space without stretching or squashing anything, so lengths and angles are preserved. Its inverse is simply its transpose — Q^T Q = I — which makes it a joy to compute with.