Dot Product Calculator

Your details

Components mode enters x, y, z values. Magnitudes + Angle reverse-solves a·b = |a||b|cosθ.
Choose whether your vectors live in two or three dimensions.
Leave at 0 if you are working in 2D.
Leave at 0 if you are working in 2D.
Dot product a·bPositive - acute
12
Angle between vectors68.56°
Angle (radians)1.1966rad
Magnitude |a|3.7417
Magnitude |b|8.775
Scalar projection of a onto b1.3675
Scalar projection of b onto a3.2071
Vector projection a onto b (x)0.6234
Vector projection a onto b (y)-0.7792
Vector projection a onto b (z)0.9351
RelationshipAcute angle (pointing similarly)
Orthogonal (perpendicular)?No
Magnitude |a|3.7417
Magnitude |b|8.775
Scalar proj a onto b1.3675

a·b = 12, angle = 68.56°, vectors are not orthogonal.

  • The dot product is 12: positive, so the vectors share a general direction (angle less than 90°).
  • Geometrically, a·b = |a||b|cosθ: it equals the length of a times the component of b along a.
  • The angle between them is 68.56°, found by arccos(a·b / (|a||b|)).
  • The scalar projection of a onto b is 1.3675: the signed length of a's shadow on b.

Next stepA dot product of exactly zero confirms orthogonality. Use the scalar projection to decompose forces or check how much one vector aligns with another.

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