1. Vectors a and b are not orthogonal, but b and c are orthogonal. We know this from their inner products, which we can calculate as follows:

2. In order for A to be an orthonormal basis, we require that the column vectors are orthogonal and have a magnitude of one. For the two column vectors to be orthogonal, we require that their inner product is zero:

We next check that the column vectors have a magnitude of one:

Both vectors are normal; therefore, the solution:

makes A an orthonormal basis.

3. In order for B to be an orthonormal basis, we require that the column vectors are orthogonal and have a magnitude of one. For the two column vectors to be orthogonal, we require that their inner product is zero:

Using the fact that the magnitude of the first column vector must be one: