Lesson Worksheet: The Perpendicular Distance between Points and Planes Mathematics

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In this worksheet, we will practice calculating the perpendicular distance between a plane and a point, between a plane and a straight line parallel to it, and between two parallel planes using a formula.

Q1:

Find the distance between the point (2,−1,3) and the plane r⋅⟨−2,2,1⟩=3.

  • A1length unit
  • B13length units
  • C23length units
  • D2length units

Q2:

Find the distance between the point (−5,−8,−6) and the plane −2𝑥+𝑦+2𝑧=7.

  • A17125 length units
  • B179 length units
  • C173 length units
  • D17 length units

Q3:

Find, to the nearest two decimal places, the distance between the point (3,2,3) and the plane 2(𝑥−2)+3(𝑦−3)+(𝑧−1)=0.

Q4:

Which of the following is the distance of the point (3,−4,5) from the plane 4𝑥21+8𝑦21+5𝑧21=1 rounded to two decimal places?

  • A1.56
  • B0.39
  • C6.10
  • D3.88
  • E0.64

Q5:

Find the perpendicular distance between the line r=⟨1,2,4⟩+𝑡⟨−2,1,4⟩ and the plane r⟨2,0,1⟩=1.

  • A10
  • B√5
  • C5
  • D√10
  • E2√5

Q6:

Find the distance between the line 𝑥−12=𝑦−24=𝑧+52 and the plane 3𝑥−2𝑦+𝑧=−2. Give your answer to one decimal place.

Q7:

Find the distance between the two planes −𝑥−2𝑦−2𝑧=−2 and −2𝑥−4𝑦−4𝑧=3.

  • A710length unit
  • B76length units
  • C25length unit
  • D23length unit

Q8:

Find, to the nearest two decimal places, the distance between the two planes 2(𝑥−2)+(𝑦−3)+3(𝑧−1)=0 and r⋅⟨4,2,6⟩=12.

Q9:

Find, to the nearest hundredth, the distance between the two planes 𝑥+2𝑦+4𝑧=4 and 2𝑥13+4𝑦13+8𝑧13=1.

Q10:

Find, to the nearest hundredth, the distance between the two planes ⟨6,3,6⟩⋅=11r and 𝑥3+𝑦6+𝑧3=1.

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