Null Vector
There are several meanings of "null vector" in mathematics.
1. The most common meaning of null vector is the 
-dimensional vector 
of length 0. i.e., the vector with 
components, each of which is 0 (Jeffreys and Jeffreys 1988,
p. 64).
2. When applied to a matrix 
, a null vector is a nonzero vector 
with the property that 
.
3. When applied to a vector space 
with an associated quadratic form 
, a null vector is a nonzero element 
of 
for which 
.
4. When applied to a geometric product satisfying the contraction rule 
for 
an element of an 
-vector space, a null vector is a value of 
such that 
but 
(Sommer 2001, pp. 5-6).
5. When applied to a vector, a null vector is a nonzero vector 
such that for a given vector 
, the dot product
satisfies 
.
(This use may be slightly nonstandard, but appears for example in the Wolfram
Language's FindIntegerNullVector
function.)
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References
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.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Null Vector." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NullVector.html