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Equation of Any Plane Passing Through the Line of Intersection of Planes

Equation of plane passing through the line of intersection of planes 87690.png and 87699.png is 87705.png91804.png, where λ R.
Equation of plane passing through the line of intersection of planes A­1x + B1y + C1zd1 = 0 and A2x + B2y + C2z d2 = 0 is (A­1x + B1y + C1zd1) + λ(A2x + B2y + C2zd2) = 0, where λ R.


  • The foot of perpendicular of point P(αβγ) on the plane ax + by + cz + d = 0 is (ar + αbr + βcr + γ ), where 91073.png
  • The image of point P(αβγ) in the planeax + by + cz + d = 0 is (ar + αbr + βcr + γ), where 91069.png
  • Line of intersection of planes 91065.png = d2 is along the vector 91060.png
  • Condition for 91056.png to lie in the plane ax + by + cz + d = 0 are al +bm + cn = 0 and ax1 + by1 + cz1 + d = 0
  • Let ax + by + cz + d = 0 be the plane then the points (x1y1z1) and (x2y2z2) lie on the same side or opposite side according as 91132.png or < 0

Properties of tetrahedron

If OABC is a tetrahedron as shown in the figure, where 91808.png. Then we have
  1. Volume of tetrahedron = 87773.png.
  2. Center of the sphere is the centroid of the tetrahedron.
  3. The angle between any two plane faces of a regular tetrahedron is 87780.png.
  4. Angle between the any edge and a face not containing the angle is 87786.png (for regular tehrahedron).
  5. Any two edges of regular tehrahedron are perpendicular to each other.
  6. The distance of any vertex from the opposite face of regular tetrahedron is 87792.pngk; k being the length of any edge.

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