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Worksheet: The Equation of a Plane in 3D in Different Forms

Q1:

Write the normal form of the plane P containing the point Q = 6 , 2 , 0 and perpendicular to the vector n = 2 , 6 , 4 .

  • A 2 𝑥 + 6 𝑦 + 4 𝑧 2 4 = 0
  • B 6 𝑥 2 𝑦 = 0
  • C 6 𝑥 2 𝑦 2 4 = 0
  • D 2 𝑥 + 6 𝑦 + 4 𝑧 = 0
  • E 2 𝑥 + 6 𝑦 + 4 𝑧 + 4 = 0

Q2:

Determine the Cartesian equation of the plane on which the projection of the origin is the point ( 1 , 4 , 5 ) .

  • A 𝑥 + 4 𝑦 + 5 𝑧 + 4 2 = 0
  • B 𝑥 + 4 𝑦 + 5 𝑧 = 0
  • C 𝑥 4 𝑦 5 𝑧 = 0
  • D 𝑥 + 4 𝑦 + 5 𝑧 4 2 = 0
  • E 𝑥 4 𝑦 5 𝑧 4 2 = 0

Q3:

The equation of a plane has general form . What is its vector form?

  • A
  • B
  • C
  • D

Q4:

Which of the following is the equation of the plane through the points ( 1 , 2 , 4 ) , ( 1 , 1 , 2 ) , and ( 2 , 2 , 1 ) ?

  • A 7 𝑥 2 𝑦 + 3 𝑧 1 5 = 0
  • B ( 𝑥 7 ) + 2 ( 𝑦 + 2 ) + 4 ( 𝑧 3 ) = 0
  • C 𝑥 + 2 𝑦 + 4 𝑧 1 5 = 0
  • D 7 ( 𝑥 1 ) 2 ( 𝑦 2 ) + 3 ( 𝑧 4 ) = 0

Q5:

Which of the following is the equation of the plane through the points ( 1 , 3 , 2 ) , ( 0 , 4 , 2 ) , and ( 3 , 4 , 2 ) ?

  • A 4 𝑥 4 𝑦 3 𝑧 + 1 0 = 0
  • B ( 𝑥 4 ) + 3 ( 𝑦 + 4 ) 2 ( 𝑧 + 3 ) = 0
  • C 𝑥 + 3 𝑦 2 𝑧 + 1 0 = 0
  • D 4 ( 𝑥 + 1 ) 4 ( 𝑦 3 ) 3 ( 𝑧 + 2 ) = 0

Q6:

Find the equation of the straight line that passes through the point and intersects the straight line orthogonally.

  • A
  • B
  • C
  • D

Q7:

Find the general equation of the plane which passes through the point ( 3 , 8 , 7 ) and contains the 𝑥 -axis.

  • A 8 𝑥 7 𝑦 = 0
  • B 7 𝑥 + 8 𝑧 = 0
  • C 3 𝑥 7 𝑦 + 8 𝑧 = 0
  • D 7 𝑦 + 8 𝑧 = 0
  • E 3 𝑥 8 𝑦 7 𝑧 = 0

Q8:

Find the general form of the equation of the plane passing through the point ( 4 , 1 , 1 ) and parallel to the plane 5 𝑥 + 6 𝑦 7 𝑧 = 0 .

  • A 4 𝑥 𝑦 + 𝑧 + 7 = 0
  • B 4 𝑥 𝑦 + 𝑧 = 0
  • C 9 𝑥 + 5 𝑦 6 𝑧 + 7 = 0
  • D 5 𝑥 + 6 𝑦 7 𝑧 + 7 = 0
  • E 5 𝑥 + 6 𝑦 7 𝑧 7 = 0