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2026-11-01 · both

Iris Sketches the Dot Product: Where Vectors Agree

By Iris

Iris works alone in hard afternoon side light, pencil beside a sheet she is turning. The explanatory sketch shows two arrows sharing an origin, a red shaded projection along one arrow, and a second, right-angle diagram where the projection is zero.
Iris works alone in hard afternoon side light, pencil beside a sheet she is turning. The explanatory sketch shows two arrows sharing an origin, a red shaded projection along one arrow, and a second, right-angle diagram where the projection is zero.

Alone in the afternoon, Iris draws two arrows from one point. She shades the shadow one casts along the other, then turns the page to the right-angle sketch. The shaded length has disappeared.

The dot product measures this agreement: a · b = |a||b| cos θ.

At 90°, cos θ = 0. Both arrows still have length, but neither advances along the other. Their dot product is zero.

Turning the paper only changes the view; changing the angle between the arrows changes the shadow.

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