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Equation of a plane passing through the line of intersection of two given planes

**Equation
of a plane passing through the line of intersection of two given planes**

The vector equation of a plane which passes through the line of
intersection of the planes

Consider the equation

The equation (3) represents a plane. Hence (1) represents a
plane.

Let _{1} be the position vector of any
point on the line of intersection of the plane. Then 1 satisfies both the equations * *â‹… 1 = *d*1 and * *â‹… 2 = *d*2 . So, we have

By (4) and (5), 1 satisfies (1). So, any point
on the line of intersection lies on the plane (1). This proves that the plane
(1) passes through the line of intersection.

The **cartesian equation **of a plane which passes through the line of
intersection of the planes *a*_{1} *x *+ *b*_{1}*y *+ *c*_{1}*z
*= *d*_{1} and *a*_{2}*x *+ *b*_{2}*y
*+ *c*_{2}*z *= *d*_{2} is given by

( *a*1 *x *+ *b*1*y *+ *c*1*z *= *d*1) Î» ( *a*2*x *+ *b*2*y *+ *c*2*z *= *d*2) = 0

Find the equation of the plane passing through the intersection
of the planes * *â‹… (*i *+ *j *+
*k *)+1 = 0 and * *â‹… (2*i -* 3*j *+ 5*k *) = 2 and
the point (âˆ’1, 2,1) .

We know that the vector equation of a plane passing through the
line of intersection of the planes

Since this plane passes through the point (âˆ’1, 2,1) , we get *Î»
*=3/5 , and hence the required equation of the plane is 11*x *âˆ’ 4 *y *+
20*z *= 1 .

Find the equation of the plane passing through the intersection of
the planes 2*x *+ 3*y *âˆ’ *z *+ 7 = 0 and *x *+ *y *âˆ’
2*z *+ 5 = 0 and is perpendicular to the plane *x *+ *y *âˆ’ 3*z
*âˆ’ 5 = 0 .

The equation of the plane passing through the intersection of the
planes 2*x *+ 3*y *âˆ’ *z *+ 7 = 0 and *x *+ *y *âˆ’ 2*z *+ 5 = 0 is (2*x *+ 3*y
*âˆ’ *z *+ 7) + *Î» *( *x *+ *y *âˆ’ 2*z *+ 5) = 0 or

(2 + *Î» *) *x *+ (3 + *Î» *) *y *+ (âˆ’1âˆ’ 2*Î»
*) *z *+ (7 + 5*Î» *) = 0

since this plane is perpendicular to the given plane *x *+ *y
*âˆ’ 3*z *âˆ’ 5 = 0 , the normals of these two planes are perpendicular to
each other. Therefore, we have

(1)(2 + *Î» *) + (1)(3 + *Î» *) + (âˆ’3)(âˆ’1âˆ’ 2*Î» *) *z
*= 0

which implies that *Î» *= âˆ’1 .Thus the required equation of
the plane is

(2*x *+ 3*y *âˆ’ *z *+ 7) âˆ’ ( *x *+ *y *âˆ’
2*z *+ 5) = 0 â‡’ *x *+ 2 *y *+
*z *+ 2 = 0 .

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Equation of a plane passing through the line of intersection of two given planes |

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