Find the vector and the cartesian equations of the lines that passes through the origin and `(5, 2, 3)`.

Explanation

Let the position vectors of the points (0,0,0) and (5,-2,3) be `veca` and ` vec b` respectively.
` :. veca = 0hati+0hatj+0hatk` and `vecb=5hati-2hatj+3hatk`
We know that vector equation of a line passing through two points whose position vectors are `veca` and `vecb` is `vecr = veca+ lambda(vecb-veca)` . Therefore,
`vecr = 0+lambda(5hati-2hatj+3hatk-0)`
`rArr vecr = lambda(5hati-2hatj+3hatk) "......."(1)`
In equation(1), put `vecr = xhati+yhatj+zhatk`,
`xhati+yhatj+zhatk=lambda(5hati-2hatj+3hatk)`
`rArr xhati+yhatj+zhatk=5lambdahati-2lambdahatj+3lambdahatk`
Equating on the coefficients of ` hati, hatj` and `hatk` on both sides,
`x = 5lambda, y = - 2lambda` and `z = 3lambda`
`:. x/5 = (y)/(-2) = z/3 = lambda`
Which is the cartesian equation of required line.