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Intuition

The product of

• the length of \(\vec{u}\) and
• the length of \(\vec{v}\) projected onto \(\vec{u}\).

Its extrema are \(\pm \|\vec{u}\| \|\vec{v}\|\), and it

equals	\(+\vert\vec{u}\vert\vert\vec{v}\vert\)	when \(\vec{u}\) points in the same direction of \(\vec{v}\),
	\(0\)	when \(\vec{u}\) is orthogonal to \(\vec{v}\), and
	\(-\vert\vec{u}\vert\vert\vec{v}\vert\)	when \(\vec{u}\) points in the opposite direction of \(\vec{v}\).