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

Q2:

Find the distance from the point 𝑄 = ( 0 , 2 , 0 ) to the plane 𝑃 5 𝑥 + 2 𝑦 7 𝑧 + 1 = 0 . Give the result correct to three decimal places.

Q3:

Determine the length of the perpendicular from a point 𝐴 ( 0 , 0 ) to the line 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 = 0 .

  • A | 𝑐 | 𝑎 × 2 + 𝑏 × 2
  • B | 𝑐 | 𝑎 + 𝑏
  • C 2 | 𝑐 | 𝑎 + 𝑏 2 2
  • D | 𝑐 | 𝑎 + 𝑏 2 2
  • E | 𝑐 | 2 𝑎 × 2 + 𝑏 × 2

Q4:

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

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

Q5:

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

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

Q6:

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

Q7:

Find, correct to two decimal places, the distance 𝑑 from the point 𝑃 = ( 1 , 1 , 1 ) to the line 𝐿 𝑥 = 2 2 𝑡 , 𝑦 = 4 𝑡 , 𝑧 = 7 + 𝑡 : .

Q8:

Find, correct to two decimal places, the distance from the point 𝑃 = ( 0 , 0 , 0 ) to the line 𝐿 𝑥 = 3 + 2 𝑡 , 𝑦 = 4 + 3 𝑡 , 𝑧 = 5 + 4 𝑡 .

Q9:

Find the length of the perpendicular from the point ( 5 , 7 ) to the straight line 𝑟 = ( 7 , 6 ) + 𝑠 ( 5 , 7 ) .

  • A 8 9 7 4
  • B 1 9 7 4 7 4
  • C 8 9 3 6
  • D 8 9 7 4 7 4

Q10:

Find the length of the perpendicular from the origin to the straight line r = 7 , 9 + 𝑠 5 , 5 .

  • A 8 5
  • B 2
  • C 8 1 0
  • D 8 2

Q11:

How far from the 𝑦 𝑧 -plane is the point ( 1 6 , 1 3 , 2 0 ) ?

Q12:

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

  • A 1 7 9 length units
  • B 1 7 1 2 5 length units
  • C17 length units
  • D 1 7 3 length units

Q13:

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

Q14:

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

Q15:

Find the distance between the point ( 4 , 2 , 2 ) and the plane 𝑟 ( 2 , 2 , 1 ) = 4 .

  • A 1 0 3 length units
  • B 2 3 length units
  • C 1 0 9 length units
  • D 2 length units

Q16:

Find the distance between the point ( 3 , 2 , 4 ) and the plane 𝑟 ( 4 , 2 , 4 ) = 1 6 .

  • A 4 3 length units
  • B 2 3 length units
  • C 2 9 length units
  • D 4 length units

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