Question
Show that the line 
lies in the plane 
Solution
The equation of given plane is
or
...(1)
The equation of given line is
...(2)
Now line (2) passes through the point (2, -3, 5), With position vector
and is parallel to the vector 
Now plane (1) passes through the point with position vector
if
i.e. if (2) (-3) + (-3) (-1) + (5) (1) = 2
i.e. if 2 = 2, which is true
Now,
is a vector normal to the plane (1). It will be perpendicular to the line (2) if it is perpendicular 
i.e. if (–3) (1) + (– 1) (– 1) + (1) (2) = 0
i.e. if –3 + 1+ 2 = 0
i.e. if 0 = 0, which is true
∴ line (2) lies in plane (1).
or
...(1)The equation of given line is
...(2)Now line (2) passes through the point (2, -3, 5), With position vector
and is parallel to the vector Now plane (1) passes through the point with position vector
if
i.e. if (2) (-3) + (-3) (-1) + (5) (1) = 2
i.e. if 2 = 2, which is true
Now,
is a vector normal to the plane (1). It will be perpendicular to the line (2) if it is perpendicular i.e. if (–3) (1) + (– 1) (– 1) + (1) (2) = 0
i.e. if –3 + 1+ 2 = 0
i.e. if 0 = 0, which is true
∴ line (2) lies in plane (1).
