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 = \((P_x+Q_x+R_x)\hat i+(P_y+Q_y+R_y)\hat j+(P_z+Q_z+R_z)\hat k\)
Sx = Px + Qx + Rx,
Sy = Pz + Qz + Rz and
Sz = Pz + Qz + Rz.
