Standard Parts

As in the case of hyperreal numbers, our next step is to introduce the standard part. If A is a finite hyperreal vector, the standard part of A is the real vector

st(A) = st(a₁)i + st(a₂)j + st(a₃)k.

Since each component of A is infinitely close to its standard part, A is infinitely close to its standard part. Thus

st(A) is the real vector infinitely close to A.

The standard part of an infinite hyperreal vector is undefined.

Here is a list of rules for standard parts of vectors. A and B are finite hyperreal vectors and c is a finite hyperreal number.

st(-A) = -st(A) st(A + B) = st(A) + st(B)

st(cA) = st(c)st(A) st(A · B) = st(A) · st(B)

st(A × B) = st(A) × st(B) st(|A|) = |st(A)|

As an example we prove the equation for inner products,

st(A · B) = st(a₁b₁ + a₂b₂ + a₃b₃)

= st(a₁)st(b₁) + st(a₂)st(b₂) + st(a₃)st(b₃)

= st(A) · st(B).

Given a nonzero hyperreal vector A, we may form its unit vector U = A/|A|. The three components of U are the direction cosines of A. As in the case of real vectors, U has length one and is parallel to A.