Mathematics

Mathematics

Problem

Find the planes passing through the intersection of planes r⋅(2i^−3j^+4k^)=1r \cdot (2\hat{i} - 3\hat{j} + 4\hat{k}) = 1 and r⋅(i^−j^)+4=0r \cdot (\hat{i} - \hat{j}) + 4 = 0 and perpendicular to planes r⋅(2i^−j^+k^)=−8r \cdot (2\hat{i} - \hat{j} + \hat{k}) = -8.
Intuitive explanation

When two planes intersect, they form a line, and infinitely many planes contain that line—they form a family. The key insight is to use the linear combination method: any plane through the intersection can be written as P1+λP2=0P_1 + \lambda P_2 = 0 for some parameter λ\lambda. Then apply the perpendicularity condition (normal vectors are perpendicular) to find which member of this family satisfies your constraint.

Intuitive explanation

When two planes intersect, they form a line, and infinitely many planes contain that line—they form a family. The key insight is to use the linear combination method: any plane through the intersection can be written as P1+λP2=0P_1 + \lambda P_2 = 0 for some parameter λ\lambda. Then apply the perpendicularity condition (normal vectors are perpendicular) to find which member of this family satisfies your constraint.

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