The length of a vector is also known as its magnitude, and is computed using Pythagoras' Theorem.

For 2D Vectors this is given as follows:

For 3D Vectors this is given as follows:

For higher dimensional vectors we have the generalised formula:

Just as the absolute value of a real number (or rational number, etc) can be computed by taking the (positive) square root of the square of the number, the magnitude of a vector is the square root of the scalar product with itself:

MatrixMultiplication Quaternion Vectors	(none)			(none)
↓	↓		↓	↓
AbsoluteValue ScalarProduct			UnitVector
			↓
		⇌	You are here MagnitudeOfAVector
			↓
			PythagorasTheorem RationalNumber RealNumber SquareRoot Vectors
↓	↓		↓	↓
UnitVector	ComplexNumber PolarRepresentationOfAComplexNumber			Adding2DVectors AddingVectors AlgebraicNumber CategoryMetaTopic CauchySequence CoDomainOfAFunction ComplexConjugate ComplexPlane ContinuedFraction CountableSet DedekindCut Denominator DifferenceOfTwoSquares DomainOfAFunction Euler Function ImageOfAFunction ImaginaryNumber Integer IrrationalNumber Numerator Pythagoras ReducingFractionsToLowestTerms RiemannSurface Root RootTwoIsIrrational TranscendentalNumber

Local neighbourhood - D3

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Magnitude Of A Vector