(i) Find the vector equation of a line passing through a point with position vector `2hat(i)-hat(j)+hat(k)`, and parallel to the line joining the points `-hat(i)+4hat(j)+hat(k)` and `hat(i)+2hat(j)+2hat(k)`. Also find the cartesian equivalent of this equation.
(ii) The cartesian equations of a line are `6x-2=3y+1=2z-2`. Find its direction ratios and also find vector equation of the line.

Answer

(i) `vec(r)=(2hat(i)-hat(j)+hat(k))+t(2hat(i)-2hat(k)+hat(k));(x-2)/(2)=(y+1)/(-2)=(z-1)/(1)`
(ii) Direction ratios are 1,2,3 and the vector equation is
`vec(r)=((1)/(3)hat(i)-(1)/(3)hat(j)+hat(k))+t((1)/(6)hat(i)+(1)/(3)hat(j)+(1)/(2)hat(k))`