Question
Find the equation of line (vector and cartesian both) which is parallel to the vector 
and which passes through the point (5, -2, 4).
Solution
We know that the equation of a straight line passing through a fixed point with position vector 
and parallel to the vector 
is
...(1)
where λ is a parameter.
Here
and 
∴ from (1), the vector equation of line is
...(2)
Now
is the position vector of (x, y, z).
∴ from (2), we get,
Comparing the coefficients of
x = 2 λ + 5, y = – (λ + 2), z = 3 λ + 4
This is the parametric form of the equation.
Again,
∴ cartesian form of the line is
and parallel to the vector is ...(1)where λ is a parameter.
Here
and ∴ from (1), the vector equation of line is
...(2)Now
is the position vector of (x, y, z).∴ from (2), we get,
Comparing the coefficients of
x = 2 λ + 5, y = – (λ + 2), z = 3 λ + 4
This is the parametric form of the equation.
Again,
∴ cartesian form of the line is
