Inner Product

We begin by addressing an earlier omission. We have discussed that we did not define any type of multiplication between vectors. While there is no operation that will have all of the properties that we believe multiplication should have, the inner product or dot product comes close. For vectors $\mathbf{x}$ and $\mathbf{y}$ in $\mathbb{R}^{n}$, we define the standard inner product as

$$\mathbf{x}\cdot \mathbf{y}= \sum_{j=1}^{n} x_{j}y_{j}.$$ (1)

The use of the dot to denote the operation has mechanic origins. In many more mathematical settings notation of the form

$$(\mathbf{x},\mathbf{y}) \equiv \mathbf{x}\cdot \mathbf{y}$$ (2)

is used. We may use either as the setting determines.

Note that this form of multiplication takes two vectors and produces a real number. Your initial thoughts on vector multiplication probably expected a vector as the result. This would be problematic. In fact, the cross product on $\mathbb{R}^{3}$ is about as close as we get. This aside, the inner product has several properties we associate with multiplication

In fact, any operation which takes two vectors into a real number and has these properties is considered an inner product.

We should also note that the inner product between two vectors can be zero. If this occurs, we say that the vectors are orthogonal or perpendicular. This terminology comes from the relationship in $\mathbb{R}^{3}$

$$(\mathbf{x},\mathbf{y}) = \Vert \mathbf{x}\Vert \Vert \mathbf{y}\Vert \cos(\theta)$$ (3)

where $\theta$ is the angle between the vectors. If $\theta = \pi/2$ radians then $\cos(\theta)=0$. Hence the vectors are perpendicular. This is harder to visualize in $\mathbb{R}^{n}$ therefore we tend to use the term orthogonal.