Question

Explain: Shifting of origin, coordinates of the point dividing a line segment externally

Answer 1

The coordinate system clearly defines the position of a vector. When we talk of the rectangular coordinates in space, we refer to the three dimensional space we live in. In order to demonstrate the position of a vector first a point is selected as the origin and generally represented by the point ‘O’. The distance of any vector is now measured form this standard point.

Consider a point P(x,y)
Let the origin be shifted to O" with coordinates (h,k) relative to old axes.
Now new P=(X,Y)
x = X+ h ; y =Y + k
$\Rightarrow x=x-h;y=y-k$

Point dividing a line segment externally

Clearly Triangle PAH is similar to Triangle PBK
Therefore, $\frac{AP}{BP}=\frac{AH}{BK}=\frac{PH}{PK}$
Therefore, $m:n=\frac{(x-x_1)}{(x-x_2)}=\frac{(y-y_1)}{(y-y_2)}$