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Worksheet: Equation of a Straight Line in Three Dimensions

Q1:

Given that 𝐿 ∢ π‘₯ + 9 βˆ’ 7 = 𝑦 βˆ’ 3 7 = 𝑧 + 8 6 1 is perpendicular to 𝐿 ∢ π‘₯ βˆ’ 2 βˆ’ 9 = 𝑦 βˆ’ 1 0 π‘˜ = 𝑧 + 3 π‘š 2 , what is 7 π‘˜ + 6 π‘š ?

Q2:

Given that the vector is parallel to the line , find .

Q3:

Given that the vector is parallel to the line , find .

Q4:

Find the parametric equations of the straight line that passes through the point 𝐴 ( βˆ’ 1 , 4 , βˆ’ 1 ) and that is parallel to the bisector of the second quadrant of the plane 𝑦 𝑧 .

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

Q5:

Suppose that lines and are parallel, what are and ?

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q6:

Suppose that lines and are parallel, what are and ?

  • A ,
  • B ,
  • C ,
  • D ,
  • E ,

Q7:

Find the Cartesian form of the equation of the straight line passing through the point ( βˆ’ 4 , 1 , 2 ) and makes equal angles with the coordinates axes.

  • A π‘₯ βˆ’ 1 βˆ’ 4 = 𝑦 βˆ’ 1 1 = 𝑧 βˆ’ 1 2
  • B π‘₯ βˆ’ 4 = 𝑦 1 = 𝑧 2
  • C π‘₯ + 4 √ 3 = 𝑦 βˆ’ 1 √ 3 = 𝑧 βˆ’ 2 3
  • D π‘₯ + 4 1 = 𝑦 βˆ’ 1 1 = 𝑧 βˆ’ 2 1

Q8:

Find the Cartesian form of the equation of the straight line passing through the point ( βˆ’ 3 , 5 , 3 ) and makes equal angles with the coordinates axes.

  • A π‘₯ βˆ’ 1 βˆ’ 3 = 𝑦 βˆ’ 1 5 = 𝑧 βˆ’ 1 3
  • B π‘₯ βˆ’ 3 = 𝑦 5 = 𝑧 3
  • C π‘₯ + 3 √ 3 = 𝑦 βˆ’ 5 √ 3 = 𝑧 βˆ’ 3 3
  • D π‘₯ + 3 1 = 𝑦 βˆ’ 5 1 = 𝑧 βˆ’ 3 1

Q9:

Given that the lines π‘₯ βˆ’ 8 3 = 𝑦 + 4 5 = 𝑧 + 6 βˆ’ 2 and π‘₯ βˆ’ 1 0 βˆ’ 5 = 𝑦 + 7 9 = 𝑧 βˆ’ 3 π‘š are perpendicular, what is π‘š ?

Q10:

Given that the lines π‘₯ + 4 βˆ’ 4 = 𝑦 + 8 βˆ’ 2 = 𝑧 βˆ’ 1 0 4 and π‘₯ + 1 βˆ’ 9 = 𝑦 βˆ’ 2 4 = 𝑧 βˆ’ 4 π‘š are perpendicular, what is π‘š ?

Q11:

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

  • A π‘₯ = 1 + 3 𝑑 , 𝑦 = βˆ’ 2 + 5 𝑑 , 𝑧 = βˆ’ 3 + 5 𝑑 , for βˆ’ ∞ < 𝑑 < ∞
  • B π‘₯ = 1 + 2 𝑑 , 𝑦 = βˆ’ 2 βˆ’ 7 𝑑 , 𝑧 = βˆ’ 3 βˆ’ 8 𝑑 , for βˆ’ ∞ < 𝑑 < ∞
  • C π‘₯ = 3 + 2 𝑑 , 𝑦 = 5 + 7 𝑑 , 𝑧 = 5 + 8 𝑑 , for βˆ’ ∞ < 𝑑 < ∞
  • D π‘₯ = 1 + 2 𝑑 , 𝑦 = βˆ’ 2 + 7 𝑑 , 𝑧 = βˆ’ 3 + 8 𝑑 , for βˆ’ ∞ < 𝑑 < ∞
  • E π‘₯ = 3 βˆ’ 2 𝑑 , 𝑦 = 5 βˆ’ 7 𝑑 , 𝑧 = 5 βˆ’ 8 𝑑 , for βˆ’ ∞ < 𝑑 < ∞

Q12:

Give the Cartesian equation of the line through point ( βˆ’ 2 , 5 , 2 ) and with direction vector ( 3 , βˆ’ 5 , βˆ’ 4 ) .

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

Q13:

Give the Cartesian equation of the line through point ( 3 , βˆ’ 2 , βˆ’ 4 ) and with direction vector ( βˆ’ 4 , βˆ’ 2 , βˆ’ 3 ) .

  • A π‘₯ + 3 βˆ’ 4 = 𝑦 + 2 βˆ’ 2 = 𝑧 + 4 βˆ’ 3
  • B π‘₯ + 4 3 = 𝑦 + 2 βˆ’ 2 = 𝑧 + 3 βˆ’ 4
  • C π‘₯ βˆ’ 4 3 = 𝑦 βˆ’ 2 βˆ’ 2 = 𝑧 βˆ’ 3 βˆ’ 4
  • D π‘₯ βˆ’ 3 βˆ’ 4 = 𝑦 + 2 βˆ’ 2 = 𝑧 + 4 βˆ’ 3

Q14:

Give the Cartesian equation of the line through point ( 8 , 3 , 7 ) and with direction vector ( βˆ’ 3 , βˆ’ 5 , βˆ’ 3 ) .

  • A π‘₯ + 8 βˆ’ 3 = 𝑦 + 3 βˆ’ 5 = 𝑧 + 7 βˆ’ 3
  • B π‘₯ + 3 8 = 𝑦 + 5 3 = 𝑧 + 3 7
  • C π‘₯ βˆ’ 3 8 = 𝑦 βˆ’ 5 3 = 𝑧 βˆ’ 3 7
  • D π‘₯ βˆ’ 8 βˆ’ 3 = 𝑦 βˆ’ 3 βˆ’ 5 = 𝑧 βˆ’ 7 βˆ’ 3

Q15:

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

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

Q16:

For what values of π‘˜ is the line 𝐿 ∢ π‘₯ βˆ’ 8 2 = 𝑦 βˆ’ 1 0 5 = 𝑧 + 1 3 1 parallel to the line 𝐿 ∢ π‘₯ βˆ’ 2 1 0 = 𝑦 βˆ’ 2 π‘˜ + 2 = 𝑧 βˆ’ 6 1 5 2 ?

Q17:

Give the equations for the 𝑧 -axis in 3-dimensional space.

  • A 𝑧 = 1
  • B π‘₯ = 0 , 𝑧 = 0
  • C 𝑧 = 0
  • D π‘₯ = 0 , 𝑦 = 0
  • E π‘₯ = 1

Q18:

Which of the following is a direction vector of the straight line π‘Ž π‘₯ + 𝑏 𝑦 + 𝑐 = 0 ?

  • A ( π‘Ž , βˆ’ 𝑏 )
  • B ( 𝑏 , π‘Ž )
  • C ( π‘Ž , 𝑏 )
  • D ( 𝑏 , βˆ’ π‘Ž )
  • E ( βˆ’ π‘Ž , βˆ’ 𝑏 )

Q19:

Which of the following is a directional vector for a line perpendicular to the π‘₯ -axis?

  • A j = ⟨ 0 , 1 ⟩
  • B i = ⟨ 1 , 0 ⟩

Q20:

Give the equations for the 𝑦 -axis in 3-dimensional space.

  • A 𝑦 = 1
  • B π‘₯ = 0 , 𝑦 = 0
  • C 𝑦 = 0
  • D π‘₯ = 0 , 𝑧 = 0
  • E 𝑧 = 1