Find the equation of the line in vector and in cartesian form that passes through the point with position vector `2 hat i- hat j+4 hat k` and is in the direction ` hat i+2 hat j- hat k` .

Explanation

Given that the lines passes through the point `veca=2hati-hatj+4hatk` and `vecb=hati+2hatj-hatk`. Therefore, equation of the line is `vecr=veca+lambdavecb` where `lambda` is a constant.
`veca=2hati-hatj+4hatk+lambda(hati+2hatj-hatk)`
Which is the required euqation of the line in vector from.
Let `vecr = x hati+yhatj+zhatk`
`:. xhati+yhatj-zhatk=(lambda+2)hati+(2lambda-1)hatj+(-lambda+4)hatk`
The cartesian equation by elimianting `lambda` is
`(x-2)/(1) = (y+1)/(2) = (z-4)/(-1)`.
Which is the required cartesian equation of the line.