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Since the square of length is a sum of squares, and squares (of real numbers) are always positive, length must always be positive.
| a | = | (a1, a2, a3 )T | = ( a12 + a22 + a32 ) >= 0
The only time the length of a 3D vector is zero is when the vector is the zero vector. In all coordinate frames the 3D zero vector is represented by:
0 = ( 0, 0, 0)T
So its length is:
| 0 | = ( 02 + 02 + 02 ) = 0
Of course, the length of the 2D zero vector is also zero, and it is the only 2D vector with zero length.