Worksheet: Parametric Equations of a Straight Line in Space

In this worksheet, we will practice finding the parametric equations of a straight line in space.

Q1:

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

  • A 𝑥 = 4 + 2 𝑡 , 𝑦 = 1 + 𝑡 , 𝑧 = 5 + 3 𝑡 , for < 𝑡 <
  • B 𝑥 = 6 + 4 𝑡 , 𝑦 = 1 , 𝑧 = 5 + 2 𝑡 , for < 𝑡 <
  • C 𝑥 = 2 6 𝑡 , 𝑦 = 1 , 𝑧 = 3 2 𝑡 , for < 𝑡 <
  • D 𝑥 = 4 6 𝑡 , 𝑦 = 1 , 𝑧 = 5 2 𝑡 , for < 𝑡 <
  • E 𝑥 = 2 + 6 𝑡 , 𝑦 = 1 + 𝑡 , 𝑧 = 3 + 2 𝑡 , for < 𝑡 <

Q2:

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

  • A 𝑥 = 8 + 8 𝐾 , 𝑦 = 6 + 7 𝐾
  • B 𝑥 = 8 6 𝐾 , 𝑦 = 8 + 7 𝐾
  • C 𝑥 = 8 + 7 𝐾 , 𝑦 = 8 6 𝐾
  • D 𝑥 = 8 + 7 𝐾 , 𝑦 = 8 + 6 𝐾

Q3:

Find the parametric equations of the straight line 3 𝑥 7 9 = 8 𝑦 3 4 = 8 6 𝑧 9 .

  • A 𝑥 = 3 7 1 3 𝑡 , 𝑦 = 8 3 + 2 𝑡 , 𝑧 = 3 4 + 2 3 𝑡
  • B 𝑥 = 3 + 7 3 𝑡 , 𝑦 = 1 2 + 3 8 𝑡 , 𝑧 = 3 2 4 3 𝑡
  • C 𝑥 = 7 3 + 3 𝑡 , 𝑦 = 3 8 1 2 𝑡 , 𝑧 = 4 3 3 2 𝑡
  • D 𝑥 = 7 3 3 𝑡 , 𝑦 = 3 8 + 1 2 𝑡 , 𝑧 = 4 3 + 3 2 𝑡

Q4:

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

  • A 𝑥 = 1 + 2 𝑡 , 𝑦 = 1 4 𝑡 , 𝑧 = 5 + 4 𝑡
  • B 𝑥 = 3 𝑡 , 𝑦 = 5 𝑡 , 𝑧 = 9 𝑡
  • C 𝑥 = 4 + 5 𝑡 , 𝑦 = 4 𝑡 , 𝑧 = 2 + 𝑡
  • D 𝑥 = 2 + 𝑡 , 𝑦 = 4 𝑡 , 𝑧 = 4 + 5 𝑡

Q5:

Determine the parametric equations of the straight line passing through the points ( 3 , 3 , 4 ) and ( 5 , 2 , 4 ) .

  • A 𝑥 = 5 + 3 𝑡 , 𝑦 = 2 + 3 𝑡 , 𝑧 = 4 + 4 𝑡
  • B 𝑥 = 5 + 8 𝑡 , 𝑦 = 2 + 5 𝑡 , 𝑧 = 4
  • C 𝑥 = 3 + 5 𝑡 , 𝑦 = 3 + 2 𝑡 , 𝑧 = 4 4 𝑡
  • D 𝑥 = 5 + 2 𝑡 , 𝑦 = 2 𝑡 , 𝑧 = 4 8 𝑡

Q6:

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

  • A 𝑥 = 7 8 𝐾 , 𝑦 = 1 0 + 4 𝐾
  • B 𝑥 = 7 1 0 𝐾 , 𝑦 = 8 + 4 𝐾
  • C 𝑥 = 7 + 4 𝐾 , 𝑦 = 8 1 0 𝐾
  • D 𝑥 = 7 + 4 𝐾 , 𝑦 = 8 + 1 0 𝐾

Q7:

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

  • A 𝑥 = 1 5 𝐾 , 𝑦 = 2 𝐾
  • B 𝑥 = 1 2 𝐾 , 𝑦 = 5 𝐾
  • C 𝑥 = 1 𝐾 , 𝑦 = 5 2 𝐾
  • D 𝑥 = 1 𝐾 , 𝑦 = 5 + 2 𝐾

Q8:

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

  • A 𝑥 = 9 1 0 𝐾 , 𝑦 = 2 3 𝐾
  • B 𝑥 = 9 + 2 𝐾 , 𝑦 = 1 0 3 𝐾
  • C 𝑥 = 9 3 𝐾 , 𝑦 = 1 0 + 2 𝐾
  • D 𝑥 = 9 3 𝐾 , 𝑦 = 1 0 2 𝐾

Q9:

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

  • A 𝑥 = 6 3 𝐾 , 𝑦 = 6 + 1 0 𝐾
  • B 𝑥 = 6 + 6 𝐾 , 𝑦 = 3 + 1 0 𝐾
  • C 𝑥 = 6 + 1 0 𝐾 , 𝑦 = 3 + 6 𝐾
  • D 𝑥 = 6 + 1 0 𝐾 , 𝑦 = 3 6 𝐾

Q10:

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

  • A 𝑥 = 5 + 7 𝐾 , 𝑦 = 8 + 2 𝐾
  • B 𝑥 = 5 + 8 𝐾 , 𝑦 = 7 + 2 𝐾
  • C 𝑥 = 5 + 2 𝐾 , 𝑦 = 7 + 8 𝐾
  • D 𝑥 = 5 + 2 𝐾 , 𝑦 = 7 8 𝐾

Q11:

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

  • A 𝑥 = 7 1 0 𝐾 , 𝑦 = 6 𝐾
  • B 𝑥 = 7 + 6 𝐾 , 𝑦 = 1 0 𝐾
  • C 𝑥 = 7 𝐾 , 𝑦 = 1 0 + 6 𝐾
  • D 𝑥 = 7 𝐾 , 𝑦 = 1 0 6 𝐾

Q12:

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

  • A 𝑥 = 8 + 9 𝐾 , 𝑦 = 6 𝐾
  • B 𝑥 = 8 6 𝐾 , 𝑦 = 9 𝐾
  • C 𝑥 = 8 𝐾 , 𝑦 = 9 6 𝐾
  • D 𝑥 = 8 𝐾 , 𝑦 = 9 + 6 𝐾

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