Question

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$\hat j$-$\hat k$.

Answer 1

It is given that the line passes through the point with the position vector
$\vec a$=2$\hat i$-$\hat j$+4$\hat k$...(i)
$\vec b$=$\hat i$+2$\hat j$-$\hat k$....(ii)

It is known that a line through a point with position vector a→ and parallel to b→ is given by the equation,
$\vec r$=$\vec a$+λ$\vec b$
⇒ $\vec r$=2$\hat i$-$\hat j$+4$\hat k$+λ($\hat i$+2$\hat j$-$\hat k$)

This is the required equation of the line in vector form.
r→=x$\hat i$-y$\hat j$+z$\hat k$
⇒x$\hat i$-y$\hat j$+z$\hat k$=(λ+2$\hat i$+(2λ-1)$\hat j$+(-λ+4)$\hat k$

Eliminating, we obtain the cartesian form equation as $\frac{x-2}{1}$=$\frac{y+1}{2}$=$\frac{z-4}{-1}$

This is the required equation of the given line in cartesian form.