Find the vector and the cartesian equations of the line that passes through the points`(3, 2, 5), (3, 2, 6)`.

Explanation

Let `veca` and `vecb` be the position vectors of the points `(3,-2,-5)` and `(3,-2,6)`.
`:. veca = 3hati-2hatj-5hatk` and `vecb = 3hati-2hatj+6hatk`
Vector equation of a line passing through two points where position vectors are `veca` and `vecb`, is
`:. vecr=3hati-2hatj-5hatk+lambda[(3hati-2hatj+6hatk)-(3hati-2hatj-5hatk)]`
`rArr vecr=3hati-2hatj-5hatk+lambda[3hati-2hatj+6hatk-3hati+2hatj+5hatk]`
`rArr vecr=3hati-2hatj-5hatk+lambda(11hatk)"......"(1)`
Which is the vector equation of the given line.
In equation `(1)`, put `vecr = xhati+yhatj+zhatk`,
`xhati+hatj+zhatk=3hati-2hatj+(11lambda-5) hatk`
Equating coefficients of `hati, hatj` and `hatj` on both sides of the equation,
`x = 3 , y = - 2` and `z = 11 lambda - 5`
`:. (x-3)/(0) = (y+2)/(0) = (z+5)/(11)`
Which is the cartesian equation of the given line.