7.3 Representation of a Vector in a plane
Let P (x, y) be any point on the coordinate plane.
Any such vector can be broken down into two components, a
Here x and y are components of p along x-axis and y-axis respectively. This ordered pair (x , y) associated with p is known as the rectangular components of p.
Let ÐPOX = q then
which is the magnitude of p.
This concept can be extended to higher dimensions such as three dimension.
Also, note that any vector can be shifted so that its initial point A is set at 'O', then is in 'standard position'. Now = then replaces .
Thus is the standard vector in the plane for all vectors in the plane of the same magnitude and direction as .
\ OP = OM + ON
If coordinates of A are (xa, ya) and that of B are (xb, yb) then shifting A to O and making B = P we have coordinates of P are (xb - xa, yb - ya)
\ or º (xb - xa,
yb - ya)
Thus an algebraic vector is an ordered pair of real numbers that corresponds to a standard geometric vector .