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.