6.3 Angle

In addition to the length of a vector, the angle between two nonzero vectors \(\mathbf{x}\) and \(\mathbf{y}\) can also be expressed using inner products. The angle \(\theta\) is such that:

\[ cos(\theta) = \frac{\mathbf{x^\mathsf{T} y}}{\sqrt{\mathbf{x^\mathsf{T} x}} \hspace{1mm} \sqrt{\mathbf{y^\mathsf{T} y}}} \]

or equivalently

\[ cos(\theta) = \frac{\mathbf{x^\mathsf{T} y}}{\| \mathbf{x} \| \hspace{1mm} \| \mathbf{y} \|} \]

Rearranging some terms, we can reexpress the formula of the inner product as:

\[ \mathbf{x^\mathsf{T} y} = \| \mathbf{x} \| \hspace{1mm} \| \mathbf{y} \| \hspace{1mm} cos(\theta) \]

The angle between weight and height in \(\mathbf{X}\) is such that:

\[ cos(\theta) = \frac{20594}{228.2192 \times 91.2688} = 0.9887 \]