[b]Definition: [/b]Let [math]a^{\rightarrow}=\left(a_1,b_1\right)[/math] and [math]b^{\rightarrow}=\left(a_2,b_2\right)[/math] be any two vectors. Then their difference, denoted by [math]a^{\rightarrow}-b^{\rightarrow}[/math], is defined as [math]a^{\rightarrow}-b^{\rightarrow}[/math] = [math]\left(a_1,b_1\right)-\left(a_2,b_2\right)[/math] = [math]\left(a_{1-}a_2,b_{1-}b_2\right)[/math]. For example, (6,8)-(10,9) = (6-10,8-9) = (-4,-1) [br][math]a^{\rightarrow}=\left(a_1,b_1\right)[/math] and [math]b^{\rightarrow}=\left(a_2,b_2\right)[/math][math]a^{\rightarrow}=\left(a_1,b_1\right)[/math] and [math]b^{\rightarrow}=\left(a_2,b_2\right)[/math][math]a^{\rightarrow}-b^{\rightarrow}[/math]

[b]Properties of addition of vectors: [/b](i) [math]a^{\rightarrow}+b^{\rightarrow}=b^{\rightarrow}+a^{\rightarrow}[/math] (commutative law)[br](ii) [math]a^{\rightarrow}+\left(b^{\rightarrow}+c^{\rightarrow}\right)=\left(a^{\rightarrow}+b^{\rightarrow}\right)+c^{\rightarrow}[/math] (Associative law)[br](iii) [math]m\left(a^{\rightarrow}+b^{\rightarrow}\right)=ma^{\rightarrow}+mb^{\rightarrow}[/math] (Scalar multiplication distributes over the addition of vectors)

[b]Multiplication of a vector by a scalar: [/b]Let [math]a^{\rightarrow}=\left(a_1,a_2\right)[/math] be any vector and k be any scalar then [math]ka^{\rightarrow}=k\left(a_1,a_2\right)=\left(ka_1,ka_2\right)[/math]. This shows that the scalar multiplies both x and y components of the vectors. For example, if [math]a^{\rightarrow}=\left(4,5\right)[/math] then [math]4a^{\rightarrow}=4\left(4,5\right)[/math]= (16,20). Similarly 4(5,9) = (20,36) etc.