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The dot product

The dot product (scalar product) of two vectors gives a number, not a vector. It is used to find the angle between vectors, and to calculate work: W=F⃗⋅s⃗W=\vec F\cdot\vec s.

a⃗⋅b⃗=axbx+ayby\vec a\cdot\vec b = a_xb_x + a_yb_ydot product, component form
a⃗⋅b⃗=∣a⃗∣∣b⃗∣cos⁡θ\vec a\cdot\vec b = |\vec a||\vec b|\cos\thetadot product, geometric form

Symbols

θ\thetaangle between the vectors°

Example

a⃗=(3,4)\vec a=(3,4) and b⃗=(4,−3)\vec b=(4,-3): a⃗⋅b⃗=12−12=0\vec a\cdot\vec b=12-12=0, so the vectors are perpendicular.

The dot product is zero exactly when the vectors are perpendicular.
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Part of Foundations of Mathematics: Vectors.