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

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In this worksheet, we will practice finding the equation of a plane in different forms like the general, vector, and parametric forms.

Q1:

Find the general equation of the plane 𝑥 = 4 + 7 𝑡 + 4 𝑡 , 𝑦 = − 3 − 4 𝑡 , 𝑧 = 1 + 3 𝑡 .

  • A 𝑥 + 1 2 𝑦 − 2 8 𝑧 = 0
  • B 𝑥 − 1 2 𝑦 + 2 8 𝑧 − 1 6 = 0
  • C 𝑥 + 4 𝑦 + 7 𝑧 + 1 6 = 0
  • D 3 𝑥 + 3 𝑦 − 7 𝑧 + 4 = 0
  • E 1 2 𝑥 + 4 𝑦 + 7 𝑧 − 4 3 = 0

Q2:

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

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

Q3:

Which of the following points lies in the plane 3 ( 𝑥 + 4 ) − 2 ( 𝑦 + 1 ) − 7 ( 𝑧 − 6 ) = 0 ?

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

Q4:

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

Q5:

Write, in intercept form, the equation of the plane 1 6 𝑥 + 2 𝑦 + 8 𝑧 − 1 6 = 0 .

  • A 𝑥 1 6 + 𝑦 2 + 𝑧 8 = 1
  • B 𝑥 1 + 𝑦 8 + 𝑧 2 = 1 6
  • C 𝑥 1 6 + 𝑦 2 + 𝑧 8 = 1 6
  • D 𝑥 1 + 𝑦 8 + 𝑧 2 = 1
  • E 𝑥 1 6 + 𝑦 1 6 + 𝑧 1 6 = 1

Q6:

Find the equation of the plane 𝑥 𝑦 .

  • A 𝑥 + 𝑦 = 𝑧
  • B 𝑥 + 𝑦 = 0
  • C 𝑥 = 𝑦
  • D 𝑧 = 0
  • E 𝑧 − 𝑥 𝑦 = 0

Q7:

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

  • A − 5 𝑥 + 5 𝑦 + 9 𝑧 − 5 = 0
  • B 5 𝑥 − 7 𝑦 − 3 𝑧 − 8 7 = 0
  • C 5 𝑥 − 7 𝑦 − 3 𝑧 − 5 = 0
  • D 5 𝑥 − 7 𝑦 − 3 𝑧 + 8 7 = 0
  • E − 5 𝑥 + 5 𝑦 + 9 𝑧 + 8 7 = 0

Q8:

A plane passes through ( − 2 , − 2 , 3 ) and has normal ⟨ − 4 , 1 , − 4 ⟩ . Give its equation in vector form.

  • A r = − 6
  • B ⟨ − 4 , 1 , − 4 ⟩ ⋅ = ⟨ − 2 , − 2 , 3 ⟩ r
  • C r = ⟨ − 4 , 1 , − 4 ⟩
  • D ⟨ − 4 , 1 , − 4 ⟩ ⋅ = − 6 r

Q9:

Which of the following does the equation − 7 𝑥 − 2 𝑧 = 0 represent in three-dimensional space?

  • Aa plane containing the 𝑧 -axis
  • Ba plane containing the 𝑥 -axis
  • Ca straight line whose direction ratios are ( − 7 , 0 , − 2 )
  • Da plane containing the 𝑦 -axis

Q10:

Determine the general form of the equation for a plane in which the two straight lines 𝐿 ∶ 𝑥 + 8 − 7 = 𝑦 + 7 − 5 = 𝑧 + 5 3 and 𝐿 ∶ 𝑥 + 8 4 = 𝑦 + 7 3 = 𝑧 + 5 4 lie.

  • A − 7 𝑥 − 5 𝑦 + 3 𝑧 − 7 6 = 0
  • B − 2 9 𝑥 − 4 0 𝑦 + 𝑧 − 4 3 = 0
  • C 4 𝑥 + 3 𝑦 + 4 𝑧 + 1 4 6 = 0
  • D − 2 9 𝑥 + 4 0 𝑦 − 𝑧 + 4 3 = 0

Q11:

Determine the Cartesian equation of the straight line passing through the point ( − 2 , 9 , 2 ) that is perpendicular to the plane 5 𝑥 − 6 𝑦 − 6 𝑧 − 1 1 = 0 .

  • A 𝑥 − 5 − 2 = 𝑦 + 6 9 = 𝑧 + 6 2
  • B 𝑥 − 2 5 = 𝑦 + 9 − 6 = 𝑧 + 2 − 6
  • C 𝑥 + 5 − 2 = 𝑦 − 6 9 = 𝑧 − 6 2
  • D 𝑥 + 2 5 = 𝑦 − 9 − 6 = 𝑧 − 2 − 6

Q12:

To which of the following planes is the straight line 𝑥 − 2 4 = 𝑦 + 7 − 3 = 𝑧 + 9 6 perpendicular?

  • A 2 𝑥 − 7 𝑦 − 9 𝑧 = 0
  • B 4 𝑥 − 1 4 𝑦 − 1 8 𝑧 + 1 9 = 0
  • C 4 𝑥 + 3 𝑦 + 6 𝑧 = − 1 9
  • D 1 2 𝑥 − 9 𝑦 + 1 8 𝑧 − 1 9 = 0

Q13:

Find the general equation of the plane which passes through the two points 𝐴 ( 8 , − 7 , − 2 ) and 𝐵 ( 1 , − 4 , − 1 ) , given that the distance from the 𝑥 -intercept to the origin is equal to the distance from the 𝑦 -intercept to the origin.

  • A 4 𝑥 + 4 𝑦 + 𝑧 + 7 = 0
  • B − 7 𝑥 − 7 𝑦 − 7 4 𝑧 − 1 = 0
  • C − 7 4 𝑥 − 7 4 𝑦 − 7 𝑧 + 1 = 0
  • D 𝑥 + 𝑦 + 4 𝑧 + 7 = 0

Q14:

Determine the general equation of the plane that contains the straight line 𝑥 + 2 7 = 𝑦 − 6 5 = 𝑧 + 9 5 and that is perpendicular to the plane − 𝑥 + 𝑦 − 2 𝑧 = 2 .

  • A 7 𝑥 + 5 𝑦 + 5 𝑧 + 2 9 = 0
  • B 5 𝑥 + 3 𝑦 − 4 𝑧 − 4 4 = 0
  • C 7 𝑥 + 5 𝑦 + 5 𝑧 − 2 6 = 0
  • D 5 𝑥 − 3 𝑦 − 4 𝑧 − 8 = 0
  • E − 2 𝑥 + 6 𝑦 − 9 𝑧 + 1 2 = 0

Q15:

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

  • A 𝑎 𝑥 + 𝑏 𝑦 + 𝑑 = 0
  • B 𝑎 𝑦 + 𝑐 𝑧 + 𝑑 = 𝑥
  • C 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0
  • D 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0
  • E 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 𝑥

Q16:

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

  • A 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 𝑥
  • B 𝑎 𝑥 + 𝑐 𝑧 + 𝑑 = 0
  • C 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0
  • D 𝑎 𝑥 + 𝑏 𝑦 + 𝑑 = 0
  • E 𝑎 𝑥 + 𝑏 𝑦 + 𝑑 = 𝑧

Q17:

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

  • A 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 𝑥
  • B 𝑎 𝑥 + 𝑏 𝑦 + 𝑑 = 0
  • C 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0
  • D 𝑎 𝑥 + 𝑐 𝑧 + 𝑑 = 0
  • E 𝑎 𝑥 + 𝑐 𝑧 + 𝑑 = 𝑦

Q18:

Write, in normal form, the equation of the plane containing the point and perpendicular to the vector .

  • A
  • B
  • C
  • D
  • E

Q19:

Find the equation, in vector form, of the plane passing through the points ( 1 , 2 , 2 ) , ( 3 , 1 , − 4 ) , and ( 0 , 3 , 3 ) .

  • A ( 5 , 4 , 1 ) ⋅ = ⟨ 1 , 2 , 2 ⟩ r
  • B r = 1 5
  • C r = ⟨ 5 , 4 , 1 ⟩
  • D ⟨ 5 , 4 , 1 ⟩ ⋅ = 1 5 r

Q20:

Find the vector form of the equation of the plane containing the two straight lines r i j k i j k = ( − − 3 ) + 𝑡 ( 3 + 3 + 4 ) and r i j k i j k = ( − − 2 − 3 ) + 𝑡 ( − − 2 − 4 ) .

  • A ⟨ 2 0 , 1 6 , 9 ⟩ ⋅ = − 2 3 r
  • B ⟨ 4 , − 4 , 3 ⟩ ⋅ = − 1 r
  • C ⟨ 4 , − 8 , 3 ⟩ ⋅ = − 3 r
  • D ⟨ 4 , − 8 , 3 ⟩ ⋅ = 3 r

Q21:

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

  • A 𝑥 − 𝑦 − 𝑧 − 5 = 0
  • B 𝑥 − 𝑦 + 𝑧 + 5 = 0
  • C 𝑥 + 𝑦 − 𝑧 + 5 = 0
  • D 𝑥 + 𝑦 + 𝑧 − 5 = 0

Q22:

Given that ⃖ ⃗ 𝐴 𝐵 is parallel to the plane 8 𝑥 − 5 𝑦 − 2 𝑧 − 5 = 0 , where the coordinates of 𝐴 and 𝐵 are ( − 4 , 3 , 𝑚 ) and ( − 3 , − 3 , 𝑛 ) , respectively, find the value of ( 𝑛 − 𝑚 ) .

Q23:

Find the general equation of the plane which passes through the point ( 2 , 8 , 1 ) and is perpendicular to the two planes − 6 𝑥 − 4 𝑦 + 6 𝑧 = − 5 and 5 𝑥 + 3 𝑦 − 6 𝑧 = 3 .

  • A − 3 𝑥 − 2 𝑦 + 3 𝑧 + 1 9 = 0
  • B 3 𝑥 + 3 𝑦 + 𝑧 − 3 1 = 0
  • C 5 𝑥 + 3 𝑦 − 6 𝑧 − 2 8 = 0
  • D 3 𝑥 − 3 𝑦 + 𝑧 + 1 7 = 0
  • E 2 𝑥 + 8 𝑦 + 𝑧 + 7 8 = 0

Q24:

Find the Cartesian equation of the plane ( 𝑥 , 𝑦 , 𝑧 ) = ( − 7 , − 5 , − 3 ) + 𝑡 ( − 3 , − 8 , 1 ) + 𝑡 ( 2 , 1 , 3 ) , where 𝑡 and 𝑡 are parameters.

  • A − 3 𝑥 − 8 𝑦 + 𝑧 − 5 8 = 0
  • B 𝑥 − 7 𝑦 − 4 𝑧 + 3 0 = 0
  • C 2 𝑥 + 𝑦 + 3 𝑧 + 2 8 = 0
  • D 2 5 𝑥 − 1 1 𝑦 − 1 3 𝑧 + 8 1 = 0
  • E − 7 𝑥 − 5 𝑦 − 3 𝑧 + 1 1 = 0

Q25:

Write, in normal form, the equation of the plane containing ( − 3 , 1 , − 3 ) , ( 4 , − 4 , 3 ) , and ( 0 , 0 , 1 ) .

  • A − 1 4 𝑥 − 1 0 𝑦 + 8 𝑧 + 8 = 0
  • B − 3 𝑥 + 𝑦 − 3 𝑧 − 8 = 0
  • C − 3 𝑥 + 𝑦 − 3 𝑧 + 8 = 0
  • D − 1 4 𝑥 − 1 0 𝑦 + 8 𝑧 − 8 = 0
  • E − 1 4 𝑥 − 1 0 𝑦 + 8 𝑧 + 5 6 = 0

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