Two-dimensional vectors are nothing but ordered pairs of real numbers. Usually, vectors are denoted by special letters like \( u, v, \) and \( w \) and their components by \( a, b, c, d, e, f \) and so on. For example, the vector \( u \), where its first component is \( r \) and its second component is \( s \) is displayed as follows: $$u = (r,s)$$

The set of all two-dimensional vectors called the two-dimensional space, is denoted by \( \mathbb{R}^2.\)

Examples of two-dimensional (real) vectors include

$$(1,2), (-1,3), (4,-2), (\sqrt{2}, \pi).$$

#### Addition of two-dimensional vectors

If \( u = (a , b) \) and \( v = (c , d) \), then their addition \( u + v \) is defined as follows: $$ u + v = (a , b) + (c , d) = $$ $$ (a+c ,~b+d) $$ This means that the first components of the vectors are added to each other and the result is placed in the first component and this rule works for the second component also.

For example, if \( u = (-1 , 3) \) and \( v = (4 , -2) \), then their addition \( u + v \) is calculated as follows: $$ u + v = (-1 , 3) + (4 , -2) =$$ $$ (-1+4 , 3 + (-2)) = (3 , 1). $$

#### Subtraction of two-dimensional vectors

Note that if \(u = (a , b) \) and \( v = (c , d) \), then the subtraction of \( v \) from \( u \) is defined as follows: $$ u – v = (a , b) – (c , d) = $$ $$ (a – c , b – d). $$ This means that the first components of the vector \( v \) is subtracted from the first component of the vector \( u \) and the result is placed in the first component and this rule works for the second component also.

For example, if \( u = (-1 , 3 \pi) \) and \( v = (4 , -2\pi) \), then \( u – v \) is computed as follows:

$$ u – v = (-1 , 3 \pi) – (4 , -2 \pi) = $$ $$(-1-4, 3 \pi – (-2 \pi)) = (-5 , 5 \pi). $$

**Remark 1**. The addition of vectors has the following properties:

- \( (u + v) + w = u + (v + w) \) (associativity of vector addition).
- \( u + \vec{0} = \vec{0} + u \) (neutral element of vector addition). Note that the zero vector in two-dimensional space is \( \vec{0} = (0,0) \).
- \( u + ( -u) = – u + u = \vec{0} \) (existence of the additive inverse). Note that in the two-dimensional space, the additive inverse of \( u = (a,b) \) is \( -u = ( -a,-b) \).
- \( u + v = v + u \) (commutativity of vector addition).

Technical comment: In other words, the set of all two-dimensional real vectors together with addition is an Abelian group. Note that complex numbers are, in fact, two-dimensional real vectors and their addition and subtraction are calculated exactly in the same way.

**Exercise**. The subtraction of vectors is neither commutative nor associative. Prove this by finding suitable examples.