Dot Product from Magnitudes and Included Angle
Enter your known values, leave one input blank, and solves for the missing one. Try different units for next level excitement!
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This is the dot product seen from the geometry side rather than the component side: length times length times the cosine of the angle between. The two views are the same number, which is exactly why the dot product is useful — you can compute it cheaply from coordinates and then read it as a statement about angles. With |a| = 6, |b| = 5 and 60° between them, a·b = 6 × 5 × 0.5 = 15. Push the angle to 90° and the product collapses to zero; push past it and the product turns negative.
The everyday consequence is physical: work is a dot product, so a porter carrying a suitcase horizontally does no work against gravity at all, the force being straight up and the motion straight along. The trap is assuming that θ is any angle in your diagram; it must be the angle between the two vectors when they are placed tail to tail. Nose-to-tail sketches quietly hand you the supplement, and cos flips sign the moment you cross 90°.
- = Dot product
- = Magnitude of a
- = Magnitude of b
- = Angle between a and b
- Dot product — Scalar Projection of One Vector onto Another, Dot Product of Two 2D Vectors (Components)
- Magnitude of a — Cross Product Magnitude, Resultant of Two Vectors at an Angle
- Magnitude of b — Cross Product Magnitude, Resultant of Two Vectors at an Angle
- Angle between a and b — Cross Product Magnitude, Resultant of Two Vectors at an Angle