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Lesson: Parametric Equations of a Straight Line in Space

In this lesson, we will learn how to find the parametric equations of a straight line in space.

Worksheet • 12 Questions

Q1:

Write the equation of the straight line 𝐿 passing through the points 𝑃 = ( 4 , 1 , 5 ) 1 and 𝑃 = ( βˆ’ 2 , 1 , 3 ) 2 in parametric form.

  • A π‘₯ = 4 βˆ’ 6 𝑑 , 𝑦 = 1 , 𝑧 = 5 βˆ’ 2 𝑑 , for βˆ’ ∞ < 𝑑 < ∞
  • B π‘₯ = βˆ’ 2 + 6 𝑑 , 𝑦 = 1 + 𝑑 , 𝑧 = 3 + 2 𝑑 , for βˆ’ ∞ < 𝑑 < ∞
  • C π‘₯ = 6 + 4 𝑑 , 𝑦 = 1 , 𝑧 = 5 + 2 𝑑 , for βˆ’ ∞ < 𝑑 < ∞
  • D π‘₯ = βˆ’ 2 βˆ’ 6 𝑑 , 𝑦 = 1 , 𝑧 = 3 βˆ’ 2 𝑑 , for βˆ’ ∞ < 𝑑 < ∞
  • E π‘₯ = 4 + 2 𝑑 , 𝑦 = 1 + 𝑑 , 𝑧 = 5 + 3 𝑑 , for βˆ’ ∞ < 𝑑 < ∞

Q2:

Which of the following are the parametric equations of the line through point 𝐴 ( βˆ’ 8 , 8 ) with a direction perpendicular to vector ⃑ 𝑒 = ( βˆ’ 6 , 7 ) ?

  • A π‘₯ = βˆ’ 8 + 7 𝐾 , 𝑦 = 8 + 6 𝐾
  • B π‘₯ = βˆ’ 8 βˆ’ 6 𝐾 , 𝑦 = 8 + 7 𝐾
  • C π‘₯ = βˆ’ 8 + 7 𝐾 , 𝑦 = 8 βˆ’ 6 𝐾
  • D π‘₯ = βˆ’ 8 + 8 𝐾 , 𝑦 = βˆ’ 6 + 7 𝐾

Q3:

Which of the following are the parametric equations of the line through point 𝐴 ( 7 , βˆ’ 8 ) with a direction perpendicular to vector ⃑ 𝑒 = ( βˆ’ 1 0 , 4 ) ?

  • A π‘₯ = 7 + 4 𝐾 , 𝑦 = βˆ’ 8 + 1 0 𝐾
  • B π‘₯ = 7 βˆ’ 1 0 𝐾 , 𝑦 = βˆ’ 8 + 4 𝐾
  • C π‘₯ = 7 + 4 𝐾 , 𝑦 = βˆ’ 8 βˆ’ 1 0 𝐾
  • D π‘₯ = 7 βˆ’ 8 𝐾 , 𝑦 = βˆ’ 1 0 + 4 𝐾

Q4:

Which of the following are the parametric equations of the line through point 𝐴 ( βˆ’ 1 , βˆ’ 5 ) with a direction perpendicular to vector ⃑ 𝑒 = ( βˆ’ 2 , βˆ’ 1 ) ?

  • A π‘₯ = βˆ’ 1 βˆ’ 𝐾 , 𝑦 = βˆ’ 5 + 2 𝐾
  • B π‘₯ = βˆ’ 1 βˆ’ 2 𝐾 , 𝑦 = βˆ’ 5 βˆ’ 𝐾
  • C π‘₯ = βˆ’ 1 βˆ’ 𝐾 , 𝑦 = βˆ’ 5 βˆ’ 2 𝐾
  • D π‘₯ = βˆ’ 1 βˆ’ 5 𝐾 , 𝑦 = βˆ’ 2 βˆ’ 𝐾

Q5:

Which of the following are the parametric equations of the line through point 𝐴 ( βˆ’ 9 , βˆ’ 1 0 ) with a direction perpendicular to vector ⃑ 𝑒 = ( 2 , βˆ’ 3 ) ?

  • A π‘₯ = βˆ’ 9 βˆ’ 3 𝐾 , 𝑦 = βˆ’ 1 0 βˆ’ 2 𝐾
  • B π‘₯ = βˆ’ 9 + 2 𝐾 , 𝑦 = βˆ’ 1 0 βˆ’ 3 𝐾
  • C π‘₯ = βˆ’ 9 βˆ’ 3 𝐾 , 𝑦 = βˆ’ 1 0 + 2 𝐾
  • D π‘₯ = βˆ’ 9 βˆ’ 1 0 𝐾 , 𝑦 = 2 βˆ’ 3 𝐾

Q6:

Which of the following are the parametric equations of the line through point 𝐴 ( βˆ’ 6 , βˆ’ 3 ) with a direction perpendicular to vector ⃑ 𝑒 = ( 6 , 1 0 ) ?

  • A π‘₯ = βˆ’ 6 + 1 0 𝐾 , 𝑦 = βˆ’ 3 βˆ’ 6 𝐾
  • B π‘₯ = βˆ’ 6 + 6 𝐾 , 𝑦 = βˆ’ 3 + 1 0 𝐾
  • C π‘₯ = βˆ’ 6 + 1 0 𝐾 , 𝑦 = βˆ’ 3 + 6 𝐾
  • D π‘₯ = βˆ’ 6 βˆ’ 3 𝐾 , 𝑦 = 6 + 1 0 𝐾

Q7:

Which of the following are the parametric equations of the line through point 𝐴 ( βˆ’ 5 , 7 ) with a direction perpendicular to vector ⃑ 𝑒 = ( 8 , 2 ) ?

  • A π‘₯ = βˆ’ 5 + 2 𝐾 , 𝑦 = 7 βˆ’ 8 𝐾
  • B π‘₯ = βˆ’ 5 + 8 𝐾 , 𝑦 = 7 + 2 𝐾
  • C π‘₯ = βˆ’ 5 + 2 𝐾 , 𝑦 = 7 + 8 𝐾
  • D π‘₯ = βˆ’ 5 + 7 𝐾 , 𝑦 = 8 + 2 𝐾

Q8:

Which of the following are the parametric equations of the line through point 𝐴 ( 7 , βˆ’ 1 0 ) with a direction perpendicular to vector ⃑ 𝑒 = ( 6 , βˆ’ 1 ) ?

  • A π‘₯ = 7 βˆ’ 𝐾 , 𝑦 = βˆ’ 1 0 βˆ’ 6 𝐾
  • B π‘₯ = 7 + 6 𝐾 , 𝑦 = βˆ’ 1 0 βˆ’ 𝐾
  • C π‘₯ = 7 βˆ’ 𝐾 , 𝑦 = βˆ’ 1 0 + 6 𝐾
  • D π‘₯ = 7 βˆ’ 1 0 𝐾 , 𝑦 = 6 βˆ’ 𝐾

Q9:

Which of the following are the parametric equations of the line through point 𝐴 ( 8 , 9 ) with a direction perpendicular to vector ⃑ 𝑒 = ( βˆ’ 6 , βˆ’ 1 ) ?

  • A π‘₯ = 8 βˆ’ 𝐾 , 𝑦 = 9 + 6 𝐾
  • B π‘₯ = 8 βˆ’ 6 𝐾 , 𝑦 = 9 βˆ’ 𝐾
  • C π‘₯ = 8 βˆ’ 𝐾 , 𝑦 = 9 βˆ’ 6 𝐾
  • D π‘₯ = 8 + 9 𝐾 , 𝑦 = βˆ’ 6 βˆ’ 𝐾

Q10:

Which of the following are the parametric equations of the line through point 𝐴 ( βˆ’ 1 0 , 2 ) with a direction perpendicular to vector ⃑ 𝑒 = ( βˆ’ 8 , 8 ) ?

  • A π‘₯ = βˆ’ 1 0 + 8 𝐾 , 𝑦 = 2 + 8 𝐾
  • B π‘₯ = βˆ’ 1 0 βˆ’ 8 𝐾 , 𝑦 = 2 + 8 𝐾
  • C π‘₯ = βˆ’ 1 0 + 8 𝐾 , 𝑦 = 2 βˆ’ 8 𝐾
  • D π‘₯ = βˆ’ 1 0 + 2 𝐾 , 𝑦 = βˆ’ 8 + 8 𝐾

Q11:

Give the parametric equation of the line on point ( 2 , βˆ’ 4 , 4 ) , with direction vector ( 1 , βˆ’ 1 , 5 ) .

  • A π‘₯ = 2 + 𝑑 , 𝑦 = βˆ’ 4 βˆ’ 𝑑 , 𝑧 = 4 + 5 𝑑
  • B π‘₯ = 3 𝑑 , 𝑦 = βˆ’ 5 𝑑 , 𝑧 = 9 𝑑
  • C π‘₯ = 4 + 5 𝑑 , 𝑦 = βˆ’ 4 βˆ’ 𝑑 , 𝑧 = 2 + 𝑑
  • D π‘₯ = 1 + 2 𝑑 , 𝑦 = βˆ’ 1 βˆ’ 4 𝑑 , 𝑧 = 5 + 4 𝑑

Q12:

Find the parametric equations of the straight line 3 π‘₯ βˆ’ 7 βˆ’ 9 = 8 𝑦 βˆ’ 3 4 = βˆ’ 8 βˆ’ 6 𝑧 βˆ’ 9 .

  • A π‘₯ = 7 3 βˆ’ 3 𝑑 , 𝑦 = 3 8 + 1 2 𝑑 , 𝑧 = βˆ’ 4 3 + 3 2 𝑑
  • B π‘₯ = βˆ’ 3 + 7 3 𝑑 , 𝑦 = 1 2 + 3 8 𝑑 , 𝑧 = 3 2 βˆ’ 4 3 𝑑
  • C π‘₯ = βˆ’ 7 3 + 3 𝑑 , 𝑦 = βˆ’ 3 8 βˆ’ 1 2 𝑑 , 𝑧 = 4 3 βˆ’ 3 2 𝑑
  • D π‘₯ = 3 7 βˆ’ 1 3 𝑑 , 𝑦 = 8 3 + 2 𝑑 , 𝑧 = βˆ’ 3 4 + 2 3 𝑑
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