**Vectors addition and subtraction in terms of components**

A vector quantity can be resolved by parallelogram law into two or more vectors in different directions. *The process of resolving a vector into two or more vectors is called resolution of a vector or vector resolution. Each resolving vector is called component of the original vector.*

Resolution into perpendicular components:

Let R be the vector along OB, so that OB = R. Now taking OB as diagonal let us draw the rectangle OABC [Fig. 1]. Here the components P and Q are perpendicular to one another, i.e.,

α + β = 90^{0}

then, Sin (α + β) = Sin 90^{0} = 1 and;

Sin β = Sin (90^{0} – α) = Cos α

According to law at trainee at trigonometry we get from the triangle OAB.

P/Cos θ = Q/Sin θ = R/Sin 90^{0}

so, P = R Cos θ and Q = R sin α … …. ….. (1)

The two components P and Q are called the perpendicular components of the resultant R. P is called the horizontal component and Q is called tangential component.

Vector forms of the two components are –

*P = R Cos α î + R Sin α î*

So, their vector addition:

* R = P + Q =*** R Cos α î + R Sin α î** … …. ….. (ii)

and vector subtraction will be the vector subtraction of P and Q.

Example: Suppose P is acting along the arm AB and is acting along the arm BC of the triangle [Fig. b). Then their vector addition R = P + Q, may be represented by the third arm AC of the triangle. Again, if P and Q are acting along AB and BC then vector subtraction R = P – Q, can be represented by the third arm CA of the triangle.