Multiplication of Vector with Real Number

Given a real number (scalar) r and a vector a=(a_1, a_2), we define a new vector r a (or r*a or a multiplied by r) by

  • r a=r (a_1,a_2)=(r*a_1, r*a_2) (multiply with r inside the parenthesis).

For example,

  • 3 (1.1, 2.3) = (3*1.1, 3*2.3)=(3.3, 6.9).

We say that ra is obtained by multiplying the vector a=(a_1, a_2) by the real number r 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 for vectors a and b

  •  -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 the vectors a and b.

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