2.3 Algebra of vectors

Vectors can be added together and multiplied by scalars. Vector addition is associative ( [link] ) and commutative ( [link] ), and vector multiplication by a sum of scalars is distributive ( [link] ). Also, scalar multiplication by a sum of vectors is distributive:

In this equation, $\alpha$ is any number (a scalar). For example, a vector antiparallel to vector $\overrightarrow{A}=A_x\widehat{i}+A_y\widehat{j}+A_z\widehat{k}$ can be expressed simply by multiplying $\overrightarrow{A}$ by the scalar $\alpha =\mathrm{-1}$ :

The generalization of the number zero to vector algebra is called the null vector    , denoted by $\overrightarrow{0}$ . All components of the null vector are zero, $\overrightarrow{0}=0\widehat{i}+0\widehat{j}+0\widehat{k}$ , so the null vector has no length and no direction.

Two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ are equal vectors    if and only if their difference is the null vector:

This vector equation means we must have simultaneously $A_x-B_x=0$ , $A_y-B_y=0$ , and $A_z-B_z=0$ . Hence, we can write $\overrightarrow{A}=\overrightarrow{B}$ if and only if the corresponding components of vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ are equal:

Two vectors are equal when their corresponding scalar components are equal.

Resolving vectors into their scalar components (i.e., finding their scalar components) and expressing them analytically in vector component form (given by [link] ) allows us to use vector algebra to find sums or differences of many vectors analytically (i.e., without using graphical methods). For example, to find the resultant of two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ , we simply add them component by component, as follows:

In this way, using [link] , scalar components of the resultant vector $\overrightarrow{R}=R_x\widehat{i}+R_y\widehat{j}+R_z\widehat{k}$ are the sums of corresponding scalar components of vectors $\overrightarrow{A}$ and $\overrightarrow{B}$ :