Howard S. Hamilton --------------------------------------------------------------------- hsh2760

Week 1 ------------------------------------------------------------------------------------- Problem 1.2.10

This document is available at http://www.cs.tamu.edu/people/hamilton/math-311/1-2-10.html

Prove Theorem 1 (Including the Statement in Parenthesis).

Theorem 1 states:

In R² the vectors:(a,b) and (b,-a) are perpendicular. (One is obtained by rotating the other through a right angle).

It can be shown that vectors (a,b) and (b,-a) are perpendicular by the use of the dot product.

(a,b) dot (b,-a) = ab-ab=0

The zero dot product indicates that (a,b) and (b,-a) are perpendicular.

Thus the theorem follows that for given vector (a,b), vector (b,-a) is perpendicular.

It can be shown that one vector can be obtained from the other via rotation by noting that the

Magnitude of each vector is equal, and only the angles are changed, which is indicative of rotation.

As an example, consider the graph of (a,b) and (b,-a) with arbitrary constants 5 and 10.

If a=5 and b=10 then the plot of that vector would be:

In the case of (a,-b) or (10,-5) the following plot occurs:

Combining these plots, we arrive at:

A visual inspection indicates that the two vectors are perpendicular and 90⁰ rotations of each other.