Applet: The dot product as projection
The dot product of the vectors $\vc{a}$ (in blue) and $\vc{b}$ (in green), when divided by the magnitude of $\vc{b}$, is the projection of $\vc{a}$ onto $\vc{b}$. This projection is illustrated by the red line segment from the tail of $\vc{b}$ to the projection of the head of $\vc{a}$ on $\vc{b}$. You can change the vectors $\vc{a}$ and $\vc{b}$ by dragging the points at their ends or dragging the vectors themselves. Notice how the dot product is positive for acute angles and negative for obtuse angles. The reported number does not depend on $\|\vc{b}\|$ only because we've divided through by that magnitude.
Applet file: dot_product_projection.ggb
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