Worksheet: Equation of a Straight Line in Three Dimensions

In this worksheet, we will practice finding the direction vector and the equation of a straight line in three dimensions.

Q1:

Suppose that lines 𝑟 = ( 5 , 3 , 4 ) + 𝑡 ( 3 , 1 , 𝑔 ) and 𝑥 5 = 𝑦 4 4 = 𝑧 2 4 are parallel, what are 𝑔 and ?

  • A 𝑔 = 1 , = 1 2
  • B 𝑔 = 1 2 , = 1
  • C 𝑔 = 1 , = 1 2
  • D 𝑔 = 1 , = 1 2
  • E 𝑔 = 1 2 , = 1

Q2:

For what values of 𝑘 is the line 𝐿 𝑥 8 2 = 𝑦 1 0 5 = 𝑧 + 1 3 1 parallel to the line 𝐿 𝑥 2 1 0 = 𝑦 2 𝑘 + 2 = 𝑧 6 1 5 2 ?

Q3:

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

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

Q4:

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

  • A 𝑥 1 4 = 𝑦 1 1 = 𝑧 1 2
  • B 𝑥 4 = 𝑦 1 = 𝑧 2
  • C 𝑥 + 4 3 = 𝑦 1 3 = 𝑧 2 3
  • D 𝑥 + 4 1 = 𝑦 1 1 = 𝑧 2 1

Q5:

Given that the lines 𝑥 8 3 = 𝑦 + 4 5 = 𝑧 + 6 2 and 𝑥 1 0 5 = 𝑦 + 7 9 = 𝑧 3 𝑚 are perpendicular, what is 𝑚 ?

Q6:

Given that the vector 𝐴 = ( 2 , 𝑘 , 6 ) is parallel to the line 𝑥 6 3 = 𝑦 5 6 = 𝑧 + 4 9 , find 𝑘 .

Q7:

Given that 𝐿 𝑥 + 9 7 = 𝑦 3 7 = 𝑧 + 8 6 1 is perpendicular to 𝐿 𝑥 2 9 = 𝑦 1 0 𝑘 = 𝑧 + 3 𝑚 2 , what is 7 𝑘 + 6 𝑚 ?

Q8:

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

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

Q9:

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

  • A 𝑧 = 1
  • B 𝑥 = 0 , 𝑧 = 0
  • C 𝑧 = 0
  • D 𝑥 = 0 , 𝑦 = 0
  • E 𝑥 = 1

Q10:

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

  • A 𝑦 = 1
  • B 𝑥 = 0 , 𝑦 = 0
  • C 𝑦 = 0
  • D 𝑥 = 0 , 𝑧 = 0
  • E 𝑧 = 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 ( 1 , 4 , 1 ) and that is parallel to the bisector of the second quadrant of the plane 𝑦 𝑧 .

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

Q13:

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

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

Q14:

Which of the following is a direction vector of the straight line 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 = 0 ?

  • A ( 𝑎 , 𝑏 )
  • B ( 𝑏 , 𝑎 )
  • C ( 𝑎 , 𝑏 )
  • D ( 𝑏 , 𝑎 )
  • E ( 𝑎 , 𝑏 )

Q15:

Find the equation of the line through the origin that intersects the line 𝑟 = ( 1 , 2 , 3 ) + 𝑡 ( 3 , 5 , 1 ) 1 1 orthogonally.

  • A 𝑟 = 𝑡 ( 1 9 , 3 1 , 1 3 ) 2 2
  • B 𝑟 = 𝑡 ( 1 3 , 2 4 , 1 9 ) 2 2
  • C 𝑟 = 𝑡 ( 2 3 , 3 9 , 1 ) 2 2
  • D 𝑟 = 𝑡 ( 1 , 4 , 2 3 ) 2 2

Q16:

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

  • A 𝑟 = ( 3 , 7 , 7 ) + 𝑡 ( 3 , 7 , 7 )
  • B 𝑟 = ( 0 , 5 , 7 ) + 𝑡 ( 3 , 7 , 7 )
  • C 𝑟 = ( 0 , 5 , 7 ) + 𝑡 ( 0 , 5 , 7 )
  • D 𝑟 = ( 3 , 7 , 7 ) + 𝑡 ( 0 , 5 , 7 )

Q17:

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

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

Q18:

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

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

Q19:

For what value of 𝑎 do the lines 𝑥 5 = 𝑦 2 1 = 𝑧 2 and 𝑥 1 𝑎 = 𝑦 + 2 4 = 𝑧 + 1 4 intersect?

Q20:

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

  • A 𝑥 = 1
  • B 𝑦 = 0 , 𝑥 = 0
  • C 𝑥 = 0
  • D 𝑦 = 0 , 𝑧 = 0
  • E 𝑧 = 1

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