# Vector Addition: Analytical Method

The graphical method of adding vectors helps us in visualizing the vectors and the resultant vector, but it is sometimes tedious and has limited accuracy. It is much easier to add vectors by combining their respective components. Consider two vectors A and B in x-y plane with components A_{x}, A

_{y}and B

_{x}, B

_{y}:

A = A_{x}+ A_{y}

B = B_{x}+ B_{y}

Let R be their sum. We have

R = A + B = (A_{x}+ A_{y})+(B_{x}+ B_{y}) (i)

since vectors obey the commutative and associative laws. We can arrange and regroup the vectors in Eq. (i) as is convenient to us:

R = (A_{x}+ B_{x})+ (A_{y}+ B_{y})

Since R = R_{x}+ R_{y}

We have, R_{x }= A_{x }+ B_{x }, R_{y }= A_{y }+ B_{y}.

Thus each component of the resultant vector R is the sum of the corresponding components of A and B.

In three dimensions, we have

A = A_{x}+ A_{y} + A_{z}

B = B_{x}+ B_{y} + B_{z}

R = A + B = R_{x}+ R_{y} + R_{z}

With

R_{x }= A_{x} + B_{x}

R_{y }= A_{y} + B_{y}

R_{z }= A_{z} + B_{z}

This method can be extended to addition and subtraction of any number of vectors. For example, if vectors **a, b** and **c** are given as

a = a_{x}+ a_{y} + a_{z}

b = b_{x}+ b_{y} + b_{z}

c = c_{x}+ c_{y} + c_{z}

then, a vector **T = a + b - c** has components:

T_{x} = a_{x} + b_{x} - c_{x}

T_{y} = a_{y} + b_{y} - c_{y}

T_{z} = a_{z} + b_{z} - c_{z}