Resolution of vectors- The process of splitting up a vector into two or move vectors is known as resulting of a vector. The vector into which a given vector is splitted are called components of given vector. The resoultion of a vector into two mutually perpendicular vectors is called rectangular resolution of vector in plane and the components are called rectangular components.
Figure, shows a vector OR = R in x-y plane drawn from the origin O. Let the vector makes an angle α with the x-axis and β with the y-axis. This vector is to be resolved into two components along two mutually perpendicular unit vector i and j along x-axis and y-axis, respectively.
From point R, drop perpendiculars RP and RQ on x-axis and y-axis, respectively. The length OP is called the projection of vector OR on x-axis while length OQ is the projection of vector OR on y-axis,
According to parallelogram law of vector addition,
Thus, the vector R is resolved in to two components one along OX and the other along OY, the magnitude of the component along. OY is OQ = RY = Ry cos β i.e., in terms of unit vector i and j can be given as
If the vector R is not in the x-y plane, it may have zero projections along X,Y,Z axes vector R can be resolved into three components along the x, y and z axes. If α, β and y be the angles made by the vector R with respect to x,y and z axis respectively, then it can be written as.
Where i, j, k are the unit vectors along x, y and z axes, respectively.
Two or more can be added if their components along three mutually perpendicular axes are known If a, b and c are three vectors written as follows:
