Are elements of a basis pairwise orthogonal? If $x_1,x_2,x_3$ are a basis for a real vector space $V$, then are $x_1,x_2,x_3$ pairwise orthogonal?

No. In general, $V$ needn't have any structure (an inner product) which allows you to define orthogonality anyway, so it doesn't make sense to describe vectors as orthogonal. Even in vector spaces with inner products, bases do not need to be orthogonal. For example, $(1,0)$ and $(1,1)$ are a basis of $\mathbb{R}^2$, but are not orthogonal (in the usual sense).
