What does it means "orthonormal" vector?

Question

my teacher's slides say that

"The columns [u_1, u_2, . . . , u_m] of an orthogonal matrix are orthonormal".

The slides provide a rigorous definition of orthogonal matrix but they say nothing about orthonormal vectors.

What does it means?

Answer 1

The implied meaning is that the columns are each normal (length $1$) and are orthogonal to one another. You are right that a single vector cannot be orthonormal.

Answer 2

It might be more precise to say "the set of vectors is orthonormal." Orthogonal means the inner product of two vectors is zero. If you're dealing with real, your inner product is usually the dot product- multiply corresponding entries and add them (for complex numbers, you need to take the conjugate of one of the vectors). So just see that each pair of vectors gives zero when you take the dot product. That's orthogonal. For normal, you need to show each vector has norm (length) one. If it doesn't, that's easily fixed by finding the norm of the vector and then dividing each component by that number.