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EXAMPLE 4 Using a Weighted Euclidean Inner Product
It is important to keep in mind that norm and distance depend on the inner product being used. If the inner product is changed, then
the norms and distances between vectors also change. For example, for the vectors and in with the
Euclidean inner product, we have
and
However, if we change to the weighted Euclidean inner product of Example 2,
then we obtain
and
Unit Circles and Spheres in Inner Product Spaces
If V is an inner product space, then the set of points in V that satisfy
is called the unit sphere or sometimes the unit circle in V. In and these are the points that lie 1 unit away from the origin.
EXAMPLE 5 Unusual Unit Circles in .
(a) Sketch the unit circle in an -coordinate system in using the Euclidean inner product
(b) Sketch the unit circle in an -coordinate system in using the weighted Euclidean inner product
.
Solution (a) , so the equation of the unit circle is , or, on squaring both sides,
If , then
As expected, the graph of this equation is a circle of radius 1 centered at the origin (Figure 6.1.1a).
