Exercise 3

The dot product is defined as:

\[\vec{A} \cdot \vec{B} = \left|\vec{A}\right| \left|\vec{B}\right| \cos \varphi\]

where \(\varphi\) denotes the angle between the two vectors.

If \(\vec{A} = \vec{B}\) we get

\[\vec{A} \cdot \vec{A} = \left|\vec{A}\right| \left|\vec{A}\right| \cos \varphi = \left|\vec{A}\right|^2\]

where \(\cos \varphi =1\) if \(\varphi = 0\).