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Worksheet: Distances between Points, Straight Lines, and Planes

In this worksheet, we will practice calculating the distance between a point and a straight line or between a point and a plane using a formula for each distance.

Q1:

Find, correct to two decimal places, the distance from the point 𝑃 = ( 0 , 0 , 0 ) to the line 𝐿 ∢ π‘₯ = 3 + 2 𝑑 , 𝑦 = 4 + 3 𝑑 , 𝑧 = 5 + 4 𝑑 .

Q2:

Find the distance from the point 𝑄 = ( 4 , 1 , 2 ) to the plane 𝑃 : 3 π‘₯ βˆ’ 𝑦 βˆ’ 5 𝑧 + 8 = 0 , giving your answer to two decimal places.

Q3:

Determine the length of the perpendicular from a point 𝐴 ( 0 , 0 ) to the line π‘Ž π‘₯ + 𝑏 𝑦 + 𝑐 = 0 .

  • A | 𝑐 | √ π‘Ž Γ— 2 + 𝑏 Γ— 2
  • B | 𝑐 | √ π‘Ž + 𝑏
  • C 2 | 𝑐 | √ π‘Ž + 𝑏 2 2
  • D | 𝑐 | √ π‘Ž + 𝑏 2 2
  • E | 𝑐 | 2 √ π‘Ž Γ— 2 + 𝑏 Γ— 2

Q4:

Find, correct to two decimal places, the distance 𝑑 from the point 𝑃 = ( 1 , βˆ’ 1 , βˆ’ 1 ) to the line 𝐿 π‘₯ = βˆ’ 2 βˆ’ 2 𝑑 , 𝑦 = 4 𝑑 , 𝑧 = 7 + 𝑑 : .

Q5:

Find, to one decimal place, the perpendicular distance from point ( βˆ’ 3 , βˆ’ 4 , 0 ) to the line on points ( 1 , 3 , 1 ) and ( 4 , 3 , 2 ) .

Q6:

Find, to one decimal place, the perpendicular distance from point ( βˆ’ 3 , βˆ’ 3 , 2 ) to the line on points ( βˆ’ 2 , 0 , 4 ) and ( 0 , βˆ’ 5 , 2 ) .

Q7:

Find, to one decimal place, the perpendicular distance from point ( 4 , βˆ’ 1 , 3 ) to the line on points ( 0 , βˆ’ 4 , βˆ’ 4 ) and ( βˆ’ 5 , 4 , βˆ’ 3 ) .

Q8:

Find the distance between the point and the plane .

  • A length unit
  • B length units
  • C length units
  • D length units

Q9:

Find the distance between the point and the plane .

  • A length units
  • B length units
  • C length units
  • D length units

Q10:

Find the distance between the point and the plane .

  • A length units
  • B length units
  • C length units
  • D length unit

Q11:

Find the length of the perpendicular from the origin to the straight line .

  • A
  • B
  • C
  • D

Q12:

Find the distance from the point 𝑄 = ( 0 , 2 , 0 ) to the plane 𝑃 ∢ βˆ’ 5 π‘₯ + 2 𝑦 βˆ’ 7 𝑧 + 1 = 0 . Give the result correct to three decimal places.

Q13:

Find the distance between the two planes βˆ’ π‘₯ βˆ’ 2 𝑦 βˆ’ 2 𝑧 = βˆ’ 2 and βˆ’ 2 π‘₯ βˆ’ 4 𝑦 βˆ’ 4 𝑧 = 3 .

  • A 7 1 0 length unit
  • B 2 3 length unit
  • C 2 5 length unit
  • D 7 6 length units

Q14:

Find the distance between the two planes βˆ’ 2 π‘₯ + 2 𝑦 + 𝑧 = 3 and 2 π‘₯ βˆ’ 2 𝑦 βˆ’ 𝑧 = 2 .

  • A 5 length units
  • B 1 length unit
  • C 3 length units
  • D 5 3 length units

Q15:

Find the distance between the two planes βˆ’ 2 π‘₯ βˆ’ 𝑦 + 2 𝑧 = βˆ’ 2 and βˆ’ 4 π‘₯ βˆ’ 2 𝑦 + 4 𝑧 = βˆ’ 1 .

  • A 3 2 length units
  • B 2 3 length unit
  • C 2 length units
  • D 1 2 length unit

Q16:

Find the length of the perpendicular from the point to the straight line .

  • A
  • B
  • C
  • D

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