Mathematics : Vector Algebra: Equation of line of intersection of two planes

**Equation
of line of intersection of two planes**

Let * *â‹…* *= *p *and * *â‹…* *= *q *be two non-parallel
planes. We know that * *and are perpendicular
to the given planes respectively.

So, the line of intersection of these planes is
perpendicular to both Ã— . * *and * *.
Therefore, it is parallel to the vector * *Ã— . Let

Consider the equations of two planes *a*_{1}*x *+
*b*_{1}*y *+ *c*_{1}*z *= *p and a*_{2}*x
*+ *b*_{2}*y *+ *c*_{2}*z *= *q *.
The line of intersection of the two given planes intersects atleast one of the
coordinate planes. For simplicity, we assume that the line meets the coordinate
plane *z *= 0 . Substitute z=0 and obtain the two equations *a*_{1}*x
*+ *b*_{1}*y *âˆ’ *p *= 0 and *a*_{2} *x
*+ *b*_{2}*y *âˆ’ *q *= 0 .Then by solving these
equations, we get the values of *x *and *y *as *x*_{1}
and *y*_{1} respectively.

So, ( *x*_{1} , *y*_{1} , 0) is
a point on the required line, which is parallel to *l*_{1}*i*Ë†
+ *l*_{2} Ë†*j *+ *l*_{3}*k*Ë† . So, the
equation of the line is

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Equation of line of intersection of two planes |

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