Plane - Introduction
A plane can be determined uniquely if anyone of the following is known:
(i) The normal to the plane and its distance from origin is given.
(ii) It passes through a point and is perpendicular to a given direction.
(iii) It passes through three given non collinear points.
Equation of plane in normal form
Vector Form: If r is the position vector of a point P in the plane, d is the perpendicular distance from origin and ň is the unit normal to the plane then its vector equation is given by
r. ň = d
Cartesian Form: If P(x, y, z) is a point in the plane, d is the perpendicular distance from origin and <l, m, n> are the direction cosines of ň, then the Cartesian form of the plane is given by
lx +my +nz =d
Note: If <a, b, c> are the direction ratios of the normal to the plane then the equation is ax+ by+ cz = d
Equation of a plane perpendicular to a given vector and passing through a given point
Vector Form: If a is the position vector of a given point and N is the perpendicular vector then its equation is given by
( r- a). N=0
Cartesian Form: If A(x1, y1, z1) is the given point and P(x, y, z) is a general point in the plane and A, B and C are the direction ratios of N then the Cartesian equation is given by
Equation of a plane passing through three non collinear points
Vector Form: If a, b and c are the position vectors of three points and r be any point in the plane, then the equation of the plane passing through three given points is
( r- a).[( b- a) X ( c- a)]=0
Cartesian Form: If (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) are the three given points then equation of the plane is
Intercept Form of a plane
If the plane makes intercepts a, b and c on x, y and z axes respectively then its equation in intercept form is given by
Here coordinates of A, B and C are A(a,0,0), B(0,b,0) and C(0,0,c) respectively.
Intersection of two planes
Vector Form: If r. n1=d1 and r. n2=d2 are the vector equation of two planes then equation of the plane passing through the intersection of these two planes is given by
r.( n1+λ n2)=d1+λd2
Cartesian Form: If A1x+B1y+C1z=d1 and A2x+B2y+C2z=d2 are the equations of two planes in the Cartesian form then the equation of the plane passing through the intersection of the given planes is
In general, if P1 and P2 are the equations of two planes then the equation of the plane passing through the intersection of P1 and P2 is given by
Example: Find the equation of the plane through the intersection of the planes x+y+z-6=0 and 2x+3y+4z+5=0 ant the point (1, 1, 1)
Solution: Equation of the plane passing through x+y+z-6=0 and 2x+3y+4z+5=0 is given by
(x+y+z-6) + λ (2x+3y+4z+5)=0 ……………(1)
Passes through (1, 1, 1)
(1+1+1-6) + λ (2+3+4+5) = 0
Substitute the value of λ in (1), so that the equation is
(x+y+z-6)+3/14(2x+3y+4z+5) = 0
20x+23y+26z-69=0, which is the required equation.
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