Worksheet: Distances between Points and Straight Lines or Planes

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In this worksheet, we will practice calculating the perpendicular 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∶−5𝑥+2𝑦−7𝑧+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 𝑎𝑥+𝑏𝑦+𝑐=0.

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

Q4:

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

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

Q5:

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

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

Q6:

Find the distance from the point 𝑄=(4,1,2) to the plane 𝑃: 3𝑥−𝑦−5𝑧+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 𝐿𝑥=−2−2𝑡,𝑦=4𝑡,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+2𝑡,𝑦=4+3𝑡,𝑧=5+4𝑡.

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

Q10:

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

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

Q11:

How far from the 𝑦𝑧-plane is the point (−16,−13,20)?

Q12:

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

  • A 1 7 1 2 5 length units
  • B 1 7 9 length units
  • C 1 7 3 length units
  • D17 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 1 0 9 length units
  • C 2 3 length units
  • D 2 length units

Q16:

Find the distance between the point (3,2,4) and the plane r⋅⟨−4,2,4⟩=−16.

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

Q17:

Let 𝐿 be the line through point (7,5,5) in the direction of vector (2,4,−9). Find the distance between 𝐿 and the point (2,6,6), to the nearest hundredth.

Q18:

Let 𝐿 be the line through the point (−6,8,9) that makes equal angles with the three coordinate axes. What is the distance between the point (−4,5,3) and 𝐿, to the nearest hundredth.

Q19:

Determine, to the nearest hundredth, the length of the perpendicular drawn from the point (−5,−7,−10) to the straight line 𝑥+82=𝑦−98=𝑧+7−8.

Q20:

Determine, to the nearest hundredth, the distance between the point (7,−5,−4), the straight line passing through the point (0,−2,2), and its direction ratios (−9,7,−5).

Q21:

Find the length of the perpendicular drawn from point 𝐴(−8,1,10) to the straight line r=⟨−1,2,−7⟩+𝑡⟨−9,−9,6⟩ rounded to the nearest hundredth.

Q22:

Find, to the nearest hundredth, the distance between the parallel lines 𝐿∶𝑥+79=𝑦+15=𝑧−7−6 and 𝐿∶𝑥+39=𝑦+105=𝑧+10−6.

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