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Scalar multiplication of a vector by a factor of 3 stretches the vector out. The scalar multiplications −a and 2a of a vector a. In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra [1] [2] [3] (or more generally, a module in abstract algebra [4] [5]).
Multiplication of (,,) by a scalar results in the multiplication of (,,) by the same scalar, and , and cannot be all unless , and are all zero since is nonsingular. So ( X , Y , Z ) {\displaystyle (X,Y,Z)} are a new system of homogeneous coordinates for the same point of the projective plane.
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A scalar beside a vector (either or both of which may be in parentheses) implies scalar multiplication. The two common operators, a dot and a rotated cross, are also acceptable (although the rotated cross is almost never used), but they risk confusion with dot products and cross products, which operate on two vectors.