Find the vector and Cartesian equation of the plane which passes through the point (5, 2, -4) and perpendicular to the line with direction ratios 2, 3, -1.

Given: The plane is passing through P(5, 2, -4) and perpendicular to the line having 2, 3, -1 as the direction ratios.


To find: the equation of the plane


Let the position vector of this point P be



And it is also given the plane is normal having 2, 3, -1 as the direction ratios.


Then



We know that the vector equation of a plane passing through the point and perpendicular/normal to the vector is given by



Substituting the values from eqn(i) and eqn(ii) in the above equation, we get




(by multiplying the two vectors using the formula )




is the vector equation of a required plane.


Let


Then, the above vector equation of the plane becomes,



Now multiplying the two vectors using the formula, we get




This is the Cartesian form of the equation of the required plane.


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