35.1: Inner Products

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Definition: An inner product on a vector space \(V\) (Remember that \(R^n\) is just one class of vector spaces) is a function that associates a number, denoted as \(\langle u,v \rangle\), with each pair of vectors \(u\) and \(v\) of \(V\). This function satisfies the following conditions for vectors \(u,v,w\) and scalar \(c\):

\(\langle u,v \rangle = \langle v,u \rangle\) (symmetry axiom)
\(\langle u+v,w \rangle = \langle u,w \rangle + \langle v,w \rangle\) (additive axiom)
\(\langle cu,v \rangle = c\langle v,u \rangle\) (homogeneity axiom)
\(\langle u,u \rangle \ge 0 \text{ and } \langle u,u \rangle = 0 \text{ if and only if } u = 0\) (positive definite axiom)

The dot product of \(R^n\) is an inner product. Note that we can define new inner products for \(R^n\).

Norm of a vector

Definition: Let \(V\) be an inner product space. The norm of a vector \(v\) is denoted by \(\| v \|\) and is defined by:

\[\| v \| = \sqrt{\langle v,v \rangle}. \nonumber\]

Angle between two vectors

Definition: Let \(V\) be a real inner product space. The angle \(\theta\) between two nonzero vectors \(u\) and \(v\) in \(V\) is given by:

\[\cos(\theta) = \frac{\langle u,v \rangle}{\| u \| \| v \|}. \nonumber\]

Orthogonal vectors

Definition: Let \(V\) be an inner product space. Two vectors \(u\) and \(v\) in \(V\) are orthogonal if their inner product is zero:

\[\langle u,v \rangle = 0. \nonumber\]

Distance

Definition: Let \(V\) be an inner product space. The distance between two vectors (points) \(u\) and \(v\) in \(V\) is denoted by \(d(u,v)\) and is defined by:

\[d(u,v) = \| u-v \| = \sqrt{\langle u-v, u-v \rangle} \nonumber\]

Example:

Let \(R^2\) have an inner product defined by: \(\langle (a_1,a_2),(b_1,b_2)\rangle = 2a_1b_1 + 3a_2b_2.\)

Question 1

What is the norm of (1,-2) in this space?

Question 2

What is the distance between (1,-2) and (3,2) in this space?

Question 3

What is the angle between (1,-2) and (3,2) in this space?

Question 4

Determine if (1,-2) and (3,2) are orthogonal in this space?