Let P, Q and R be represented in component from i.e.,

\(\vec P\) = \(P_x\hat i+P_y\hat j+P_z\hat k\)

\(\vec Q\) = \(Q_x\hat i+Q_y\hat j+Q_z\hat k\)

\(\vec R\)= \(R_x\hat i+R_y\hat j+R_z\hat k\)

Let \(\vec S\) be their summation vector, i.e.,

\(\vec S=\vec P+\vec Q+\vec R\)

= \((P_x\hat i+P_y\hat j+P_z\hat k)+(Q_x\hat i+Q_y\hat j+Q_z\hat k)+(R_x\hat i+R_y\hat j+R_z\hat k)\)

Addition of vectors obey the commutative as well as associative laws

∴ S = \((P_x+Q_x+R_x)\hat i+(P_y+Q_y+R_y)\hat j+(P_z+Q_z+R_z)\hat k\)

S_{x} = P_{x} + Q_{x} + R_{x,}

S_{y} = P_{z} + Q_{z} + R_{z} and

S_{z} = P_{z} + Q_{z} + R_{z}.