If the position coordinates of the points A and B are `(x_1,y_1,z_1) and (x_2,y_2,z_2)` respectively , determine the magnitude and direction of the vector `vec(AB)`.

Explanation

In Fig.2.74 , the position coordinates of A and B are `(x_1,y_1,z_1) and (x_2,y_2,z_2)` respectively.

`therefore vec(OA)=x_1hati+y_1hatj+z_1hatk and vec(OB)=x_2hati+y_2hatj+z_2hatk`
From `Delta OAB` we get,
`vec(OA)+vec(AB)=vec(OB)`
`or, vec(AB)=vec(OB)-vec(OA)=(x_2hati+y_2hatj+z_2hatk)-(x_1hati+y_1hatj+z_1hatk)`
`=(x_2-x_1)hati+(y_2-y_1)hatj+(z_2-z_1)hatk`
The magnitude of the vector `vec(AB)` is,
`AB=sqrt((x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2).`
The direction of `vec(AB)` is obtained from its direction cosines
`(x_2-x_1)/(AB),(y_2-y_1)/(AB) and (z_2-z_1)/(AB)`.