The multiple of an arbitrary vector \( u \) by a real number \( r \), denoted by \( ru \), is obtained by multiplying each component of \( u \) by the given number \( r \). In other words, if \( u = (a,b) \) is a two-dimensional vector, then $$ r u = r(a,b) = (ra, rb). $$

Also, if \( u = (a,b,c) \) is a three-dimensional vector, then $$r u = r(a,b,c) = (ra, rb,rc).$$

For example, the multiple of the vector \( u = (2,\sqrt{3}) \) by the real number \( \sqrt{3} \) is calculated as follows: $$\sqrt{3} (2,\sqrt{3}) = (2 \sqrt{3} , 3).$$

Also, the multiple of the vector \( u = (2,\sqrt{3}, 5) \) by the real number \( -1 \) is as follows: $$-1 (2,\sqrt{3} , 5) = (- 2, – \sqrt{3} , – 5) $$ i.e. its additive inverse.

### The properties of the multiple of vectors by numbers

**Remark**. The multiple of vectors by real numbers satisfies the following distributive laws:

- \( r (u+v) = ru + rv \)
- \( (r+s) u = ru + su \)

The multiple of vectors by real numbers has the following additional properties and their proof is left to the reader as an exercise.

- \( 0 u = \vec{0} \) and \( r \vec{0} = \vec{0}. \)
- The multiple of a vector \( u \) by the real number \( – 1 \), i.e., \( (-1) u \) is its additive inverse, i.e., \( – u. \)

### Parallel vectors

The geometric interpretation of the concept of the multiple of a vector by a real number is as follows:

Two vectors are, by definition, parallel if one of them is a nonzero multiple of the other one. In other words, the vectors \(u\) and \(v\) are parallel if there is a nonzero real number \(k\) such that $$ u = k v.$$

**Example**. The vectors \( (3,3) \) and \( (-2,-2) \) are parallel because if we multiply the vector \( (-2,-2) \) by \(-3/2\), we obtain the vector \( (3,3). \)

In some references, when a vector is multiplied by a real number, it is said that a vector is multiplied by a real scalar. In this direction, the term “scalar multiplication” of a real number \( r \) and a vector \( u \) refers to the product of a real number and a vector, denoted by \( r u. \)

Note that the multiple of a vector by a number is, in fact, a function of the form \( \cdot : \mathbb{R} \times V \rightarrow V \) defined by \( (r,u) \mapsto ru,\) where \(V\) is a real vector space.

**Exercise**. Show that the vector \( (1,0) \) cannot be parallel to any vector of the form \( (0,b).\)

Addition of vectors and scalar multiplication of a real number and a vector are essential concepts in linear algebra and multivariable calculus. In particular, with the help of these two concepts, linear combination and linear independence are defined.