# 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 $b$.