Q2 of 181 Page 1181

Find the vector and Cartesian equations of a plane which is at a distance of 7 units from the origin and whose normal vector from the origin is

Given :


d = 7



To Find : Equation of plane


Formulae :


1) Unit Vector :


Let be any vector


Then unit vector of is



Where,


2) Dot Product :


If are two vectors




then,



3) Equation of plane :


Equation of plane which is at a distance of 5 units from the origin and having as a unit vector normal to it is



Where,


For given normal vector



Unit vector normal to the plane is






Equation of the plane is





This is a vector equation of the plane.


Now,



= (x × 3) + (y × 5) + (z × (-6))


= 3x + 5y – 6z


Therefore equation of the plane is



This is the Cartesian equation of the plane.


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9

If O is the origin and P(1, 2, -3) be a given point, then find the equation of the plane passing through P and perpendicular to OP.

 1

Find the vector and Cartesian equations of a plane which is at a distance of 5 units from the origin and which has as the unit vector normal to it.

 3

Find the vector and Cartesian equations of a plane which is at a distance of from the origin and whose normal vector from the origin is

4

Find the vector and Cartesian equations of a plane which is at a distance of 6 units from the origin and which has a normal with direction ratios 2, -1, -2.