Equation 3-2 shows that the addition of two vectors$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{}\mathbf{and}\mathbf{}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{b}}$is commutative. Does that mean subtraction is commutative, so that$\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}\mathbf{}\mathbf{-}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{b}}\mathbf{=}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{b}}\mathbf{-}\stackrel{\mathbf{\rightharpoonup }}{\mathbf{a}}$?

The addition of two vectors $\stackrel{\rightharpoonup }{a}\mathrm{and}\stackrel{\rightharpoonup }{\mathrm{b}}$is commutative.

role="math" localid="1660887998302" $\stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{\mathrm{b}}=\stackrel{\rightharpoonup }{b}+\stackrel{\rightharpoonup }{a}$

The laws of vector addition and subtraction dictate the way in which the vectors can be added or subtracted. If the vectors are exactly in the same direction, we can apply the normal addition and subtraction laws to the vectors. But, if they are not in the same direction, we have to resolve the vectors along the unit vectors and add or subtract them.

Formula:

$$\begin{gathered} \stackrel{\rightharpoonup }{a}+\stackrel{\rightharpoonup }{\mathrm{b}}=\stackrel{\rightharpoonup }{b}+\stackrel{\rightharpoonup }{a} \\ \stackrel{\rightharpoonup }{a}-\stackrel{\rightharpoonup }{\mathrm{b}}=\left(-\stackrel{\rightharpoonup }{\mathrm{b}}\right)+\stackrel{\rightharpoonup }{a} \end{gathered}$$

If the $\stackrel{\rightharpoonup }{a}\mathrm{and}\stackrel{\rightharpoonup }{b}$are two vectors then using vector addition,

Hence, vector addition is commutative.

When two vectors are subtracted then, we have to check if,

$\stackrel{\rightharpoonup }{a}-\stackrel{\rightharpoonup }{\mathrm{b}}=-\stackrel{\rightharpoonup }{b}+\stackrel{\rightharpoonup }{a}$

From this figure, we can say that $\stackrel{\rightharpoonup }{a}\mathrm{and}\stackrel{\rightharpoonup }{b}$are not commutative, because,

$\stackrel{\rightharpoonup }{a}-\stackrel{\rightharpoonup }{\mathrm{b}}\ne -\stackrel{\rightharpoonup }{b}+\stackrel{\rightharpoonup }{a}$

Thus, it is proved that vector addition is commutative but subtraction of two vectors is not commutative.