Question

Find the Cartesian equation of the plane, passing through the line of intersection of the planes:   \[r.(2\hat i + 3\hat j - 4\hat k) + 5 = 0\]  and  \[r.(\hat i - 5\hat j + 7\hat k) + 2 = 0\] and intersecting y-axis at  \[\left( {0,3} \right)\]

Answer 1

Hint: In this problem we will approach with a direct vector approach. We will first find the equation of the plane of the intersection of two planes with a lambda form. Then as it intersects y axis at some point, we will find another condition to proceed and find the value of lambda.

Complete step-by-step answer:
We have two planes as, \[r.(2\hat i + 3\hat j - 4\hat k) + 5 = 0\] and \[r.(\hat i - 5\hat j + 7\hat k) + 2 = 0\] .
As, any plane passing through the intersection of two planes \[r.(a\hat i + b\hat j + c\hat k) + d = 0\] and \[r.(e\hat i + f\hat j + g\hat k) + h = 0\] given as, \[[r.(a\hat i + b\hat j + c\hat k) + d] + \lambda [r.(e\hat i + f\hat j + g\hat k) + h] = 0\] .
Now, Any plane passing through the intersection of the planes \[r.(2\hat i + 3\hat j - 4\hat k) + 5 = 0\] and \[r.(\hat i - 5\hat j + 7\hat k) + 2 = 0\] is,
  \[[r.(2\hat i + 3\hat j - 4\hat k) + 5] + \lambda [r.(\hat i - 5\hat j + 7\hat k) + 2] = 0\]
Now, on simplifying, we get,
 \[r[(2 + \lambda )\hat i + (3 - 5\lambda )\hat j + ( - 4 + 7\lambda )\hat k] = - 5 - 2\lambda \] ……(i)
Again, If equation (i) intersects the y-axis at \[\left( {0,3} \right)\] , then the point \[\left( {0,3} \right)\] will satisfy the equation. In other words the point \[\left( {0,3} \right)\] will be on that line.
So, now the point, \[r = 0\hat i + 0\hat j + 3\hat k\] ,
As it is on the line, the dot product with the given intersection of planes will be zero.
So, we get, \[(0\hat i + 0\hat j + 3\hat k).[(2 + \lambda )\hat i + (3 - 5\lambda )\hat j + ( - 4 + 7\lambda )\hat k] = 0\]
 \[ \Rightarrow 0(2 + \lambda ) + 0(3 - \lambda ) + 3( - 4 + 7\lambda ) = 0\]
 \[ \Rightarrow - 12 + 21\lambda = 0\]
 \[ \Rightarrow 21\lambda = 12\]
 \[ \Rightarrow \lambda = \dfrac{{12}}{{21}} = \dfrac{4}{7}\]
∴ The required plane is obtained after substituting the value of \[\lambda \]
 \[[(2 + \dfrac{4}{7})\hat i + (3 - 5.\dfrac{4}{7})\hat j + ( - 4 + 7.\dfrac{4}{7})\hat k] = - 5 - 2.\dfrac{4}{7}\]
 \[ \Rightarrow (\dfrac{{18}}{7})\hat i + (\dfrac{{21 - 20}}{7})\hat j + ( - 4 + 4)\hat k] = \dfrac{{ - 35 - 8}}{7}\]
 \[ \Rightarrow (\dfrac{{18}}{7})\hat i + (\dfrac{1}{7})\hat j = \dfrac{{ - 43}}{7}\]
Multiplying both sides by 7, we get,
 \[ \Rightarrow 18\hat i + \hat j = - 43\]
So, we have, the equation of the plane, as,
  \[18x + y = - 43\] or \[18x + y + 43 = 0\;\] which is the required equation.

Note: Planes are either parallel, or they're perpendicular, otherwise they intersect each other at some other angle. Planes are said to be parallel if the ratio equality is true, whereas they are perpendicular if the dot product of their normal vectors is 0.