Worksheet: Equation of a Straight Line in Space: Parametric Form

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

Q1:

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

  • A 𝑥 = 4 6 𝑡 , 𝑦 = 1 , 𝑧 = 5 2 𝑡 , for <𝑡<
  • B 𝑥 = 4 + 2 𝑡 , 𝑦 = 1 + 𝑡 , 𝑧 = 5 + 3 𝑡 , for <𝑡<
  • C 𝑥 = 2 + 6 𝑡 , 𝑦 = 1 + 𝑡 , 𝑧 = 3 + 2 𝑡 , for <𝑡<
  • D 𝑥 = 2 6 𝑡 , 𝑦 = 1 , 𝑧 = 3 2 𝑡 , for <𝑡<
  • E 𝑥 = 6 + 4 𝑡 , 𝑦 = 1 , 𝑧 = 5 + 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 + 7 𝐾 , 𝑦 = 8 + 6 𝐾
  • B 𝑥 = 8 + 8 𝐾 , 𝑦 = 6 + 7 𝐾
  • C 𝑥 = 8 6 𝐾 , 𝑦 = 8 + 7 𝐾
  • D 𝑥 = 8 + 7 𝐾 , 𝑦 = 8 6 𝐾

Q3:

Find the parametric equations of the straight line 3𝑥79=8𝑦34=86𝑧9.

  • A 𝑥 = 7 3 + 3 𝑡 , 𝑦 = 3 8 1 2 𝑡 , 𝑧 = 4 3 3 2 𝑡
  • B 𝑥 = 3 7 1 3 𝑡 , 𝑦 = 8 3 + 2 𝑡 , 𝑧 = 3 4 + 2 3 𝑡
  • C 𝑥 = 3 + 7 3 𝑡 , 𝑦 = 1 2 + 3 8 𝑡 , 𝑧 = 3 2 4 3 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 = 4 + 5 𝑡 , 𝑦 = 4 𝑡 , 𝑧 = 2 + 𝑡
  • C 𝑥 = 2 + 𝑡 , 𝑦 = 4 𝑡 , 𝑧 = 4 + 5 𝑡
  • D 𝑥 = 3 𝑡 , 𝑦 = 5 𝑡 , 𝑧 = 9 𝑡

Q5:

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

  • A 𝑥 = 3 + 5 𝑡 , 𝑦 = 3 + 2 𝑡 , 𝑧 = 4 4 𝑡
  • B 𝑥 = 5 + 8 𝑡 , 𝑦 = 2 + 5 𝑡 , 𝑧 = 4
  • C 𝑥 = 5 + 3 𝑡 , 𝑦 = 2 + 3 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=10,4?

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

Q8:

Which of the following are the parametric equations of the line through point 𝐴(9,10) with a direction perpendicular to vector u=2,3?

  • A 𝑥 = 9 3 𝐾 , 𝑦 = 1 0 + 2 𝐾
  • B 𝑥 = 9 + 2 𝐾 , 𝑦 = 1 0 3 𝐾
  • C 𝑥 = 9 3 𝐾 , 𝑦 = 1 0 2 𝐾
  • D 𝑥 = 9 1 0 𝐾 , 𝑦 = 2 3 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=6,10?

  • A 𝑥 = 6 + 1 0 𝐾 , 𝑦 = 3 6 𝐾
  • B 𝑥 = 6 3 𝐾 , 𝑦 = 6 + 1 0 𝐾
  • C 𝑥 = 6 + 6 𝐾 , 𝑦 = 3 + 1 0 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 + 2 𝐾 , 𝑦 = 7 + 8 𝐾
  • C 𝑥 = 5 + 8 𝐾 , 𝑦 = 7 + 2 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(7,10) with a direction perpendicular to vector u=6,1?

  • A 𝑥 = 7 + 6 𝐾 , 𝑦 = 1 0 𝐾
  • B 𝑥 = 7 𝐾 , 𝑦 = 1 0 + 6 𝐾
  • C 𝑥 = 7 𝐾 , 𝑦 = 1 0 6 𝐾
  • D 𝑥 = 7 1 0 𝐾 , 𝑦 = 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 𝐾 , 𝑦 = 9 6 𝐾
  • B 𝑥 = 8 𝐾 , 𝑦 = 9 + 6 𝐾
  • C 𝑥 = 8 + 9 𝐾 , 𝑦 = 6 𝐾
  • D 𝑥 = 8 6 𝐾 , 𝑦 = 9 𝐾

Q13:

Find the point of intersection of the lines 𝐿𝑥=7+3𝑡,𝑦=43𝑡,𝑧=75𝑡: and 𝐿𝑥=1+6𝑡,𝑦=2+𝑡,𝑧=32𝑡:.

  • A ( 7 , 4 , 7 )
  • B ( 6 , 1 , 2 )
  • C ( 7 , 3 , 7 )
  • D ( 1 , 2 , 3 )
  • E ( 3 , 3 , 5 )

Q14:

Find the point of intersection of the lines 𝑥64=𝑦+3=𝑧 and 𝑥113=𝑦146=𝑧+92.

  • A 4 , 1 1 2 , 5 2
  • Bno intersection point
  • C ( 1 1 , 1 4 , 9 )
  • D 8 3 4 , 1 1 2 , 5 2
  • E ( 6 , 3 , 0 )

Q15:

Which of the following points lies on the straight line r=3,2,1+𝑡2,1,5?

  • A ( 1 , 3 , 0 )
  • B ( 1 , 2 , 4 )
  • C ( 2 , 0 , 3 )
  • D ( 3 , 5 , 1 6 )

Q16:

The line 𝑥25=𝑦27=𝑧110 passes through the sphere 𝑥+𝑦+𝑧18𝑥+8𝑦+14𝑧+28=0. Find the length of the line segment between the two points of intersection of the line and the sphere. Give your answer to the nearest hundredth.

Q17:

A cube with side 3 sits with a vertex at the origin and three sides along the positive axes. Find the parametric equations of the main diagonal from the origin.

  • A 𝑥 = 3 + 𝑡 , 𝑦 = 3 + 𝑡 , 𝑧 = 3 + 𝑡
  • B 𝑥 = 3 , 𝑦 = 3 , 𝑧 = 3
  • C 𝑥 = 3 𝑡 , 𝑦 = 3 𝑡 , 𝑧 = 3 𝑡
  • D 𝑥 = 1 + 3 𝑡 , 𝑦 = 1 + 3 𝑡 , 𝑧 = 1 + 3 𝑡

Q18:

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

  • A 𝑥 = 3 2 𝑡 , 𝑦 = 5 7 𝑡 , 𝑧 = 5 8 𝑡 , for <𝑡<
  • B 𝑥 = 1 + 2 𝑡 , 𝑦 = 2 7 𝑡 , 𝑧 = 3 8 𝑡 , for <𝑡<
  • C 𝑥 = 3 + 2 𝑡 , 𝑦 = 5 + 7 𝑡 , 𝑧 = 5 + 8 𝑡 , for <𝑡<
  • D 𝑥 = 1 + 2 𝑡 , 𝑦 = 2 + 7 𝑡 , 𝑧 = 3 + 8 𝑡 , for <𝑡<
  • E 𝑥 = 1 + 3 𝑡 , 𝑦 = 2 + 5 𝑡 , 𝑧 = 3 + 5 𝑡 , for <𝑡<

Q19:

Give the parametric equation of the line through the origin with direction vector (5,1,4).

  • A 𝑥 = 5 , 𝑦 = 1 , 𝑧 = 4
  • B 𝑥 = 4 𝑡 , 𝑦 = 𝑡 , 𝑧 = 5 𝑡
  • C 𝑥 = 𝑡 , 𝑦 = 4 𝑡 , 𝑧 = 5 𝑡
  • D 𝑥 = 5 𝑡 , 𝑦 = 𝑡 , 𝑧 = 4 𝑡

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