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Lesson: Equation of a Straight Line in Three Dimensions

Worksheet • 20 Questions

Q1:

Suppose that lines ⃑ π‘Ÿ = ( 5 , βˆ’ 3 , 4 ) + 𝑑 ( βˆ’ 3 , βˆ’ 1 , 𝑔 ) and π‘₯ βˆ’ 5 β„Ž = 𝑦 βˆ’ 4 βˆ’ 4 = 𝑧 βˆ’ 2 4 are parallel, what are 𝑔 and β„Ž ?

  • A 𝑔 = 1 , β„Ž = βˆ’ 1 2
  • B 𝑔 = 1 2 , β„Ž = βˆ’ 1
  • C 𝑔 = βˆ’ 1 2 , β„Ž = 1
  • D 𝑔 = 1 , β„Ž = 1 2
  • E 𝑔 = βˆ’ 1 , β„Ž = βˆ’ 1 2

Q2:

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 ?

Q3:

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

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

Q4:

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

Q5:

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 ?

Q6:

Given that the vector ⃑ 𝐴 = ( 2 , π‘˜ , 6 ) is parallel to the line π‘₯ βˆ’ 6 3 = 𝑦 βˆ’ 5 6 = 𝑧 + 4 9 , find π‘˜ .

Q7:

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 ?

Q8:

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

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

Q9:

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

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

Q10:

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

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

Q11:

Which of the following is a directional vector for a line perpendicular to the -axis?

  • A
  • B

Q12:

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 , 𝑦 = 4 βˆ’ 𝑑 , 𝑧 = βˆ’ 1 + 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 + 1 2 𝑑 , 𝑦 = 4 + 1 2 𝑑 , 𝑧 = βˆ’ 1 + 1 2 𝑑

Q13:

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

Q14:

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

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

Q15:

Find the equation of the line through the origin that intersects the line ⃑ π‘Ÿ = ( βˆ’ 1 , 2 , 3 ) + 𝑑 ( 3 , βˆ’ 5 , 1 ) 1 1 orthogonally.

  • A ⃑ π‘Ÿ = 𝑑 ( βˆ’ 1 , 4 , 2 3 ) 2 2
  • B ⃑ π‘Ÿ = 𝑑 ( βˆ’ 1 3 , 2 4 , 1 9 ) 2 2
  • C ⃑ π‘Ÿ = 𝑑 ( βˆ’ 2 3 , 3 9 , βˆ’ 1 ) 2 2
  • D ⃑ π‘Ÿ = 𝑑 ( 1 9 , βˆ’ 3 1 , 1 3 ) 2 2

Q16:

Give the vector equation of the line through the point ( 3 , 7 , βˆ’ 7 ) with direction vector ( 0 , βˆ’ 5 , 7 ) .

  • A ⃑ π‘Ÿ = ( 3 , 7 , βˆ’ 7 ) + 𝑑 ( 0 , βˆ’ 5 , 7 )
  • B ⃑ π‘Ÿ = ( 0 , βˆ’ 5 , 7 ) + 𝑑 ( 3 , 7 , βˆ’ 7 )
  • C ⃑ π‘Ÿ = ( 0 , βˆ’ 5 , 7 ) + 𝑑 ( 0 , βˆ’ 5 , 7 )
  • D ⃑ π‘Ÿ = ( 3 , 7 , βˆ’ 7 ) + 𝑑 ( 3 , 7 , βˆ’ 7 )

Q17:

Find the direction vector of the straight line passing through 𝐴 ( 1 , βˆ’ 2 , 7 ) and 𝐡 ( 4 , βˆ’ 1 , 3 ) .

  • A ⃑ 𝑑 = ( 3 , 1 , βˆ’ 4 )
  • B ⃑ 𝑑 = ( βˆ’ 3 , 1 , 4 )
  • C ⃑ 𝑑 = ( 3 , βˆ’ 1 , βˆ’ 4 )
  • D ⃑ 𝑑 = ( 5 , βˆ’ 3 , 1 0 )

Q18:

Give a direction vector of the line through the origin and the point ( 6 , 6 , 1 ) .

  • A ⃑ 𝑑 = ( 6 , 6 , 1 )
  • B ⃑ 𝑑 = ( βˆ’ 6 , 6 , βˆ’ 1 )
  • C ⃑ 𝑑 = ( 6 , βˆ’ 6 , 1 )
  • D ⃑ 𝑑 = ( 0 , 0 , 0 )

Q19:

For what value of π‘Ž do the lines π‘₯ βˆ’ 5 = 𝑦 βˆ’ 2 βˆ’ 1 = 𝑧 βˆ’ 2 and π‘₯ βˆ’ 1 π‘Ž = 𝑦 + 2 4 = 𝑧 + 1 4 intersect?

Q20:

Give equations for the π‘₯ -axis in 3-dimensional space.

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