Question
Download Solution PDFDetermine the vector equation of the plane passing through the intersection of the planes
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
As we know that, if a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 represents two different planes, then equation of plane passing through the intersection of these planes is given by:
(a1x + b1y + c1z + d1) + λ × (a2x + b2y + c2z + d2) = 0.
Calculation:
Given:
Convert these planes in the cartesian form.
Put
So
⇒ x + y + z = 6 .........(1)
Similarly,
⇒ 2x + 3y + 4z = -5 .........(2)
So, the plane passing through the intersection of two given planes is:
⇒ (x + y + z – 6) + λ × (2x +3y + 4z + 5) = 0
∵ it is given that the plane passing through the intersection of two given planes also passes through the point (1, 1, 1)
⇒ The point (3, 2, 1) will satisfy equation (1)
⇒ (1 + 1 + 1 - 6) + λ × (2 + 3 + 4 + 5) = 0 ⇒ λ = 3/14.
So, by substituting the value of λ in equation (1), we get
⇒ (x + y + z – 6) + (3/14) × (2x +3y + 4z + 5) = 0
20x +23y + 26z = 69
Last updated on Jul 9, 2025
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