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