Null Vector

There are several meanings of "null vector" in mathematics.

1. The most common meaning of null vector is the n-dimensional vector 0 of length 0. i.e., the vector with n components, each of which is 0 (Jeffreys and Jeffreys 1988, p. 64).

2. When applied to a matrix A, a null vector is a nonzero vector x with the property that Ax=0.

3. When applied to a vector space X with an associated quadratic form q, a null vector is a nonzero element x of X for which q(x)=0.

4. When applied to a geometric product satisfying the contraction rule a^2=epsilon_a|a|^2 for a an element of an n-vector space, a null vector is a value of a such that a!=0 but |a|=0 (Sommer 2001, pp. 5-6).

5. When applied to a vector, a null vector is a nonzero vector a such that for a given vector x, the dot product satisfies a·x=0. (This use may be slightly nonstandard, but appears for example in the Wolfram Language's FindIntegerNullVector function.)

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Jeffreys, H. and Jeffreys, B. S. "Direction Vectors." §2.033 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 64, 1988.Sommer, G. Geometric Computing with Clifford Algebras. Springer, 2001.

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Null Vector

Cite this as:

Weisstein, Eric W. "Null Vector." From MathWorld--A Wolfram Web Resource.

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