| Geometry of the Dot Product |
Dot Product Interpreted
The dot product of 2 vectors is defined as || u || || v || cos h .
From the diagram, we see that w = v - u, and from
the Law of Cosines: || w ||² = || u ||² + || v ||² - 2|| u || || v || cos h
But, w = v - u so we have || (v - u) ||² = || u ||² + || v ||² - 2|| u || || v || cos h
Therefore, || u || || v || cos h = ½ (|| u ||² + || v ||² - || (v - u) ||²)
|| u || || v || cos h = ½ ( u1² + u2² + u3 ² + v1² + v2² + v3 ² - [(v1 - u1)² + (v2 - u2)² + (v3 - u3)²])
Which works out to be u1 v1 + u 2 v 2 + u 3 v3
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Notice how we use geometry, algebra and trig in unison to approach the problem.
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