Multiplication of Vector with Real Number

Given a real number \lambda and a vector a=(a_1,a_2)\in R^2, we definea new vector \lambda a\in R^2 by

  • \lambda a=\lambda (a_1,a_2)=(\lambda a_1,\lambda a_2).

For example,

  • 3\, (1.1,2.3)=(3.3,6.9).
We say that \lambda a is obtained by multiplying the vector a=(a_1,a_2) by the real number \lambda and call this operation multiplication of a vector by a scalar.

Below we will meet other types of multiplication connected with scalar product of vectors and vector product of vectors, both being different from multiplication of a vector by a scalar.

We define

  • -a=(-1)a=(-a_1,-a_2)
  • a-b=a+(-b).

We note that a-a=a+(-a)=(a_1-a_1,a_2-a_2)=(0,0)=0. We give an example:

The sum 0.7a-b of the multiples 0.7a and (-1)b of a and b.

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