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

Q1:

Find the Cartesian equation of the plane ( π‘₯ , 𝑦 , 𝑧 ) = ( βˆ’ 7 , βˆ’ 5 , βˆ’ 3 ) + 𝑑 ( βˆ’ 3 , βˆ’ 8 , 1 ) + 𝑑 ( 2 , 1 , 3 ) 1 2 , where 𝑑 1 and 𝑑 2 are parameters.

  • A βˆ’ 3 π‘₯ βˆ’ 8 𝑦 + 𝑧 βˆ’ 5 8 = 0
  • B π‘₯ βˆ’ 7 𝑦 βˆ’ 4 𝑧 + 3 0 = 0
  • C 2 π‘₯ + 𝑦 + 3 𝑧 + 2 8 = 0
  • D 2 5 π‘₯ βˆ’ 1 1 𝑦 βˆ’ 1 3 𝑧 + 8 1 = 0
  • E βˆ’ 7 π‘₯ βˆ’ 5 𝑦 βˆ’ 3 𝑧 + 1 1 = 0

Q2:

Which of the following points lies in the plane 3 ( π‘₯ + 4 ) βˆ’ 2 ( 𝑦 + 1 ) βˆ’ 7 ( 𝑧 βˆ’ 6 ) = 0 ?

  • A ( 7 , βˆ’ 1 , βˆ’ 1 3 )
  • B ( 3 , βˆ’ 2 , βˆ’ 7 )
  • C ( 4 , 1 , βˆ’ 6 )
  • D ( βˆ’ 4 , βˆ’ 1 , 6 )

Q3:

Find the general equation of the plane which passes through the two points 𝐴 ( 8 , βˆ’ 7 , βˆ’ 2 ) and 𝐡 ( 1 , βˆ’ 4 , βˆ’ 1 ) , given that the distance from the π‘₯ -intercept to the origin is equal to the distance from the 𝑦 -intercept to the origin.

  • A 4 π‘₯ + 4 𝑦 + 𝑧 + 7 = 0
  • B βˆ’ 7 π‘₯ βˆ’ 7 𝑦 βˆ’ 7 4 𝑧 βˆ’ 1 = 0
  • C βˆ’ 7 4 π‘₯ βˆ’ 7 4 𝑦 βˆ’ 7 𝑧 + 1 = 0
  • D π‘₯ + 𝑦 + 4 𝑧 + 7 = 0

Q4:

Find the equation, in vector form, of the plane passing through the points , , and .

  • A
  • B
  • C
  • D

Q5:

Write, in normal form, the equation of the plane P containing the point Q = ⟨ 5 , 1 , βˆ’ 2 ⟩ and perpendicular to the vector n = ⟨ 4 , βˆ’ 4 , 3 ⟩ .

  • A 4 π‘₯ βˆ’ 4 𝑦 + 3 𝑧 + 1 0 = 0
  • B 5 π‘₯ + 𝑦 βˆ’ 2 𝑧 βˆ’ 1 0 = 0
  • C 5 π‘₯ + 𝑦 βˆ’ 2 𝑧 + 1 0 = 0
  • D 4 π‘₯ βˆ’ 4 𝑦 + 3 𝑧 βˆ’ 1 0 = 0
  • E 4 π‘₯ βˆ’ 4 𝑦 + 3 𝑧 + 4 = 0

Q6:

Find the general equation of a plane that is parallel to the π‘₯ -axis.

  • A π‘Ž π‘₯ + 𝑏 𝑦 + 𝑑 = 0
  • B π‘Ž 𝑦 + 𝑐 𝑧 + 𝑑 = π‘₯
  • C π‘Ž π‘₯ + 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0
  • D 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0
  • E 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = π‘₯

Q7:

Find the vector form of the equation of the plane containing the two straight lines and .

  • A
  • B
  • C
  • D

Q8:

Determine the Cartesian equation of the straight line passing through the point ( βˆ’ 2 , 9 , 2 ) that is perpendicular to the plane 5 π‘₯ βˆ’ 6 𝑦 βˆ’ 6 𝑧 βˆ’ 1 1 = 0 .

  • A π‘₯ βˆ’ 5 βˆ’ 2 = 𝑦 + 6 9 = 𝑧 + 6 2
  • B π‘₯ βˆ’ 2 5 = 𝑦 + 9 βˆ’ 6 = 𝑧 + 2 βˆ’ 6
  • C π‘₯ + 5 βˆ’ 2 = 𝑦 βˆ’ 6 9 = 𝑧 βˆ’ 6 2
  • D π‘₯ + 2 5 = 𝑦 βˆ’ 9 βˆ’ 6 = 𝑧 βˆ’ 2 βˆ’ 6

Q9:

Determine the general equation of the plane that contains the straight line π‘₯ + 2 7 = 𝑦 βˆ’ 6 5 = 𝑧 + 9 5 and that is perpendicular to the plane βˆ’ π‘₯ + 𝑦 βˆ’ 2 𝑧 = 2 .

  • A 7 π‘₯ + 5 𝑦 + 5 𝑧 + 2 9 = 0
  • B 5 π‘₯ + 3 𝑦 βˆ’ 4 𝑧 βˆ’ 4 4 = 0
  • C 7 π‘₯ + 5 𝑦 + 5 𝑧 βˆ’ 2 6 = 0
  • D 5 π‘₯ βˆ’ 3 𝑦 βˆ’ 4 𝑧 βˆ’ 8 = 0
  • E βˆ’ 2 π‘₯ + 6 𝑦 βˆ’ 9 𝑧 + 1 2 = 0

Q10:

Which of the following is the equation of a plane that bisects the line segment between the two points ( 4 , βˆ’ 2 , βˆ’ 6 ) and ( 8 , 4 , 2 ) ?

  • A π‘₯ βˆ’ 𝑦 βˆ’ 𝑧 βˆ’ 5 = 0
  • B π‘₯ βˆ’ 𝑦 + 𝑧 + 5 = 0
  • C π‘₯ + 𝑦 βˆ’ 𝑧 + 5 = 0
  • D π‘₯ + 𝑦 + 𝑧 βˆ’ 5 = 0

Q11:

Find the equation of the plane which is perpendicular to the vector A i j k = 5 βˆ’ 7 βˆ’ 3 and passes through the point 𝐡 ( βˆ’ 5 , 5 , 9 ) .

  • A βˆ’ 5 π‘₯ + 5 𝑦 + 9 𝑧 βˆ’ 5 = 0
  • B 5 π‘₯ βˆ’ 7 𝑦 βˆ’ 3 𝑧 βˆ’ 8 7 = 0
  • C 5 π‘₯ βˆ’ 7 𝑦 βˆ’ 3 𝑧 βˆ’ 5 = 0
  • D 5 π‘₯ βˆ’ 7 𝑦 βˆ’ 3 𝑧 + 8 7 = 0
  • E βˆ’ 5 π‘₯ + 5 𝑦 + 9 𝑧 + 8 7 = 0

Q12:

Find the general equation of a plane that is parallel to the 𝑧 -axis.

  • A 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = π‘₯
  • B π‘Ž π‘₯ + 𝑐 𝑧 + 𝑑 = 0
  • C π‘Ž π‘₯ + 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0
  • D π‘Ž π‘₯ + 𝑏 𝑦 + 𝑑 = 0
  • E π‘Ž π‘₯ + 𝑏 𝑦 + 𝑑 = 𝑧

Q13:

Given that βƒ–     βƒ— 𝐴 𝐡 is parallel to the plane 8 π‘₯ βˆ’ 5 𝑦 βˆ’ 2 𝑧 βˆ’ 5 = 0 , where the coordinates of 𝐴 and 𝐡 are ( βˆ’ 4 , 3 , π‘š ) and ( βˆ’ 3 , βˆ’ 3 , 𝑛 ) , respectively, find the value of ( 𝑛 βˆ’ π‘š ) .

Q14:

Find the general equation of the plane π‘₯ = 4 + 7 𝑑 + 4 𝑑 1 2 , 𝑦 = βˆ’ 3 βˆ’ 4 𝑑 2 , 𝑧 = 1 + 3 𝑑 1 .

  • A π‘₯ + 1 2 𝑦 βˆ’ 2 8 𝑧 = 0
  • B π‘₯ βˆ’ 1 2 𝑦 + 2 8 𝑧 βˆ’ 1 6 = 0
  • C π‘₯ + 4 𝑦 + 7 𝑧 + 1 6 = 0
  • D 3 π‘₯ + 3 𝑦 βˆ’ 7 𝑧 + 4 = 0
  • E 1 2 π‘₯ + 4 𝑦 + 7 𝑧 βˆ’ 4 3 = 0

Q15:

Find the general equation of the plane which passes through the point ( 2 , 8 , 1 ) and is perpendicular to the two planes βˆ’ 6 π‘₯ βˆ’ 4 𝑦 + 6 𝑧 = βˆ’ 5 and 5 π‘₯ + 3 𝑦 βˆ’ 6 𝑧 = 3 .

  • A βˆ’ 3 π‘₯ βˆ’ 2 𝑦 + 3 𝑧 + 1 9 = 0
  • B 3 π‘₯ + 3 𝑦 + 𝑧 βˆ’ 3 1 = 0
  • C 5 π‘₯ + 3 𝑦 βˆ’ 6 𝑧 βˆ’ 2 8 = 0
  • D 3 π‘₯ βˆ’ 3 𝑦 + 𝑧 + 1 7 = 0
  • E 2 π‘₯ + 8 𝑦 + 𝑧 + 7 8 = 0

Q16:

A plane passes through and has normal . Give its equation in vector form.

  • A
  • B
  • C
  • D

Q17:

Find the equation of the plane π‘₯ 𝑦 .

  • A π‘₯ + 𝑦 = 𝑧
  • B π‘₯ + 𝑦 = 0
  • C π‘₯ = 𝑦
  • D 𝑧 = 0
  • E 𝑧 βˆ’ π‘₯ 𝑦 = 0

Q18:

Find the general equation of a plane that is parallel to the 𝑦 -axis.

  • A 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = π‘₯
  • B π‘Ž π‘₯ + 𝑏 𝑦 + 𝑑 = 0
  • C π‘Ž π‘₯ + 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0
  • D π‘Ž π‘₯ + 𝑐 𝑧 + 𝑑 = 0
  • E π‘Ž π‘₯ + 𝑐 𝑧 + 𝑑 = 𝑦

Q19:

Which of the following does the equation βˆ’ 7 π‘₯ βˆ’ 2 𝑧 = 0 represent in three-dimensional space?

  • Aa plane containing the 𝑧 -axis
  • Ba plane containing the π‘₯ -axis
  • Ca straight line whose direction ratios are ( βˆ’ 7 , 0 , βˆ’ 2 )
  • Da plane containing the 𝑦 -axis

Q20:

Which of the following does the equation 𝑦 + 𝑧 = 0 represent in three-dimensional space?

  • Aa plane containing the 𝑧 -axis
  • Ba plane containing the 𝑦 -axis
  • Ca straight line whose direction ratios are ( 0 , 1 , 1 )
  • Da plane containing the π‘₯ -axis

Q21:

Which of the following does the equation 8 π‘₯ + 5 𝑦 = 0 represent in three-dimensional space?

  • Aa plane containing the 𝑦 -axis
  • Ba plane containing the π‘₯ -axis
  • Ca straight line whose direction ratios are ( 8 , 5 , 0 )
  • Da plane containing the 𝑧 -axis

Q22:

Determine the general form of the equation for a plane in which the two straight lines 𝐿 ∢ π‘₯ + 8 βˆ’ 7 = 𝑦 + 7 βˆ’ 5 = 𝑧 + 5 3 1 and 𝐿 ∢ π‘₯ + 8 4 = 𝑦 + 7 3 = 𝑧 + 5 4 2 lie.

  • A βˆ’ 7 π‘₯ βˆ’ 5 𝑦 + 3 𝑧 βˆ’ 7 6 = 0
  • B βˆ’ 2 9 π‘₯ βˆ’ 4 0 𝑦 + 𝑧 βˆ’ 4 3 = 0
  • C 4 π‘₯ + 3 𝑦 + 4 𝑧 + 1 4 6 = 0
  • D βˆ’ 2 9 π‘₯ + 4 0 𝑦 βˆ’ 𝑧 + 4 3 = 0

Q23:

Write, in intercept form, the equation of the plane 1 6 π‘₯ + 2 𝑦 + 8 𝑧 βˆ’ 1 6 = 0 .

  • A π‘₯ 1 6 + 𝑦 2 + 𝑧 8 = 1
  • B π‘₯ 1 + 𝑦 8 + 𝑧 2 = 1 6
  • C π‘₯ 1 6 + 𝑦 2 + 𝑧 8 = 1 6
  • D π‘₯ 1 + 𝑦 8 + 𝑧 2 = 1
  • E π‘₯ 1 6 + 𝑦 1 6 + 𝑧 1 6 = 1

Q24:

In which of the following planes does the point ( 3 , βˆ’ 1 , 5 ) lie?

  • A 2 π‘₯ + 𝑦 βˆ’ 2 𝑧 + 2 3 = 0
  • B βˆ’ 4 π‘₯ βˆ’ 4 𝑦 + 2 𝑧 + 7 = 0
  • C 2 π‘₯ βˆ’ 4 𝑦 + 𝑧 + 5 = 0
  • D π‘₯ βˆ’ 2 𝑦 + 2 𝑧 βˆ’ 1 5 = 0
  • E 3 π‘₯ βˆ’ 𝑦 + 5 𝑧 = 0

Q25:

To which of the following planes is the straight line π‘₯ βˆ’ 2 4 = 𝑦 + 7 βˆ’ 3 = 𝑧 + 9 6 perpendicular?

  • A 2 π‘₯ βˆ’ 7 𝑦 βˆ’ 9 𝑧 = 0
  • B 4 π‘₯ βˆ’ 1 4 𝑦 βˆ’ 1 8 𝑧 + 1 9 = 0
  • C 4 π‘₯ + 3 𝑦 + 6 𝑧 = βˆ’ 1 9
  • D 1 2 π‘₯ βˆ’ 9 𝑦 + 1 8 𝑧 βˆ’ 1 9 = 0