Linear Algebra with Applications, 10th Edition

5.1 The Scalar Product in ${ℝ}^n$

Two vectors x and y in ${ℝ}^n$ may be regarded as $n\times 1$ matrices. We can then form the matrix product ${\mathbf{x}}^T\mathbf{y}$. This product is a $1\times 1$ matrix that may be regarded as a vector in ${ℝ}^1$ or, more simply, as a real number. The product ${\mathbf{x}}^T\mathbf{y}$ is called the scalar product of x and y. In particular, if $\mathbf{x}={\left(x_1,\mathrm{\dots },x_n\right)}^T$ and $\mathbf{y}={\left(y_1,\mathrm{\dots },y_n\right)}^T$, then

$${\mathbf{x}}^T\mathbf{y}=x_1{y}_1+x_2{y}_2+\mathrm{\dots }+x_n{y}_n$$

Example 1

If

$$\begin{array}{cc}\mathbf{\text{x}}=\left[\begin{array}{c}3\\ -2\\ 1\end{array}\right]& \begin{array}{cc}\text{and}& \mathbf{\text{y}}=\left[\begin{array}{c}4\\ 3\\ 2\end{array}\right]\end{array}\end{array}$$

then

$${\mathbf{x}}^T\mathbf{y}=\left(3,-2,1\right)\left[\begin{array}{c}4\\ 3\\ 2\end{array}\right]=3\cdot 4-2\cdot 3+1\cdot 2=8$$

The Scalar Product in ${ℝ}^2$ and ${ℝ}^3$

In order to see the geometric significance of the scalar product, let us begin by restricting our attention to ${ℝ}^2$ and ${ℝ}^3$. Vectors in ${ℝ}^2$ and ${ℝ}^3$ can be represented by directed line segments. Given a vector x in either ${ℝ}^2$ or ${ℝ}^3$, its Euclidean length can be defined in terms ...