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.

  • A𝑥=26𝑡, 𝑦=1, 𝑧=32𝑡, for <𝑡<
  • B𝑥=2+6𝑡, 𝑦=1+𝑡, 𝑧=3+2𝑡, for <𝑡<
  • C𝑥=4+2𝑡, 𝑦=1+𝑡, 𝑧=5+3𝑡, for <𝑡<
  • D𝑥=6+4𝑡, 𝑦=1, 𝑧=5+2𝑡, for <𝑡<
  • E𝑥=46𝑡, 𝑦=1, 𝑧=52𝑡, 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?

  • A𝑥=8+7𝐾, 𝑦=86𝐾
  • B𝑥=8+8𝐾, 𝑦=6+7𝐾
  • C𝑥=8+7𝐾, 𝑦=8+6𝐾
  • D𝑥=86𝐾, 𝑦=8+7𝐾

Q3:

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

  • A𝑥=733𝑡, 𝑦=38+12𝑡, 𝑧=43+32𝑡
  • B𝑥=73+3𝑡, 𝑦=3812𝑡, 𝑧=4332𝑡
  • C𝑥=3713𝑡, 𝑦=83+2𝑡, 𝑧=34+23𝑡
  • D𝑥=3+73𝑡, 𝑦=12+38𝑡, 𝑧=3243𝑡

Q4:

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

  • A𝑥=2+𝑡, 𝑦=4𝑡, 𝑧=4+5𝑡
  • B𝑥=4+5𝑡, 𝑦=4𝑡, 𝑧=2+𝑡
  • C𝑥=1+2𝑡, 𝑦=14𝑡, 𝑧=5+4𝑡
  • D𝑥=3𝑡, 𝑦=5𝑡, 𝑧=9𝑡

Q5:

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

  • A𝑥=5+2𝑡, 𝑦=2𝑡, 𝑧=48𝑡
  • B𝑥=5+3𝑡, 𝑦=2+3𝑡, 𝑧=4+4𝑡
  • C𝑥=3+5𝑡, 𝑦=3+2𝑡, 𝑧=44𝑡
  • D𝑥=5+8𝑡, 𝑦=2+5𝑡, 𝑧=4

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?

  • A𝑥=7+4𝐾, 𝑦=810𝐾
  • B𝑥=78𝐾, 𝑦=10+4𝐾
  • C𝑥=7+4𝐾, 𝑦=8+10𝐾
  • D𝑥=710𝐾, 𝑦=8+4𝐾

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?

  • A𝑥=1𝐾, 𝑦=52𝐾
  • B𝑥=15𝐾, 𝑦=2𝐾
  • C𝑥=1𝐾, 𝑦=5+2𝐾
  • D𝑥=12𝐾, 𝑦=5𝐾

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?

  • A𝑥=93𝐾, 𝑦=10+2𝐾
  • B𝑥=910𝐾, 𝑦=23𝐾
  • C𝑥=93𝐾, 𝑦=102𝐾
  • D𝑥=9+2𝐾, 𝑦=103𝐾

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?

  • A𝑥=6+10𝐾, 𝑦=3+6𝐾
  • B𝑥=63𝐾, 𝑦=6+10𝐾
  • C𝑥=6+10𝐾, 𝑦=36𝐾
  • D𝑥=6+6𝐾, 𝑦=3+10𝐾

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?

  • A𝑥=5+2𝐾, 𝑦=7+8𝐾
  • B𝑥=5+7𝐾, 𝑦=8+2𝐾
  • C𝑥=5+2𝐾, 𝑦=78𝐾
  • D𝑥=5+8𝐾, 𝑦=7+2𝐾

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?

  • A𝑥=7𝐾, 𝑦=10+6𝐾
  • B𝑥=710𝐾, 𝑦=6𝐾
  • C𝑥=7𝐾, 𝑦=106𝐾
  • D𝑥=7+6𝐾, 𝑦=10𝐾

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?

  • A𝑥=8𝐾, 𝑦=96𝐾
  • B𝑥=8+9𝐾, 𝑦=6𝐾
  • C𝑥=8𝐾, 𝑦=9+6𝐾
  • D𝑥=86𝐾, 𝑦=9𝐾

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(3,3,5)
  • E(1,2,3)

Q14:

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

  • A4,112,52
  • B834,112,52
  • C(11,14,9)
  • Dno intersection point
  • 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(3,5,16)
  • C(1,2,4)
  • D(2,0,3)

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.

  • A𝑥=3, 𝑦=3, 𝑧=3
  • B𝑥=3+𝑡, 𝑦=3+𝑡, 𝑧=3+𝑡
  • C𝑥=1+3𝑡, 𝑦=1+3𝑡, 𝑧=1+3𝑡
  • D𝑥=3𝑡, 𝑦=3𝑡, 𝑧=3𝑡

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.

  • A𝑥=1+3𝑡, 𝑦=2+5𝑡, 𝑧=3+5𝑡, for <𝑡<
  • B𝑥=1+2𝑡, 𝑦=2+7𝑡, 𝑧=3+8𝑡, for <𝑡<
  • C𝑥=3+2𝑡, 𝑦=5+7𝑡, 𝑧=5+8𝑡, for <𝑡<
  • D𝑥=32𝑡, 𝑦=57𝑡, 𝑧=58𝑡, for <𝑡<
  • E𝑥=1+2𝑡, 𝑦=27𝑡, 𝑧=38𝑡, for <𝑡<

Q19:

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

  • A𝑥=5𝑡, 𝑦=𝑡, 𝑧=4𝑡
  • B𝑥=5, 𝑦=1, 𝑧=4
  • C𝑥=4𝑡, 𝑦=𝑡, 𝑧=5𝑡
  • D𝑥=𝑡, 𝑦=4𝑡, 𝑧=5𝑡

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.