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Worksheet: Finding the Intersection between Planes

Q1:

Find the point of intersection of the planes βˆ’ 5 π‘₯ βˆ’ 2 𝑦 + 6 𝑧 βˆ’ 1 = 0 , βˆ’ 7 π‘₯ + 8 𝑦 + 𝑧 βˆ’ 6 = 0 , and π‘₯ βˆ’ 3 𝑦 + 3 𝑧 + 1 1 = 0 .

  • A ( βˆ’ 7 , 8 , 1 )
  • B ( βˆ’ 5 , βˆ’ 2 , 6 )
  • C ( 1 , βˆ’ 3 , 3 )
  • D ( βˆ’ 5 , βˆ’ 3 , βˆ’ 5 )
  • E ( βˆ’ 1 1 , 3 , 1 0 )

Q2:

Find the point of intersection of the planes 8 π‘₯ + 5 𝑦 βˆ’ 𝑧 βˆ’ 1 7 = 0 , 2 π‘₯ βˆ’ 4 𝑦 βˆ’ 7 𝑧 + 8 5 = 0 , and βˆ’ 6 π‘₯ + 𝑦 + 3 𝑧 βˆ’ 4 1 = 0 .

  • A ( 2 , βˆ’ 4 , βˆ’ 7 )
  • B ( 8 , 5 , βˆ’ 1 )
  • C ( βˆ’ 6 , 1 , 3 )
  • D ( βˆ’ 2 , 8 , 7 )
  • E ( 4 , 2 , βˆ’ 5 )

Q3:

Find the point of intersection of the planes 5 π‘₯ βˆ’ 3 𝑦 + 7 𝑧 + 5 = 0 , 7 π‘₯ + 3 𝑦 + 𝑧 βˆ’ 5 = 0 , and 2 π‘₯ βˆ’ 9 𝑦 + 2 𝑧 βˆ’ 1 6 = 0 .

  • A ( 7 , 3 , 1 )
  • B ( 5 , βˆ’ 3 , 7 )
  • C ( 2 , βˆ’ 9 , 2 )
  • D ( 2 , βˆ’ 2 , βˆ’ 3 )
  • E ( 1 4 , βˆ’ 9 , 1 0 )

Q4:

Find the point of intersection of the planes 7 π‘₯ + 8 𝑦 + 2 𝑧 + 1 3 0 = 0 , 3 π‘₯ + 7 𝑦 βˆ’ 𝑧 + 7 3 = 0 , and 4 π‘₯ βˆ’ 2 𝑦 βˆ’ 9 𝑧 βˆ’ 6 6 = 0 .

  • A ( 3 , 7 , βˆ’ 1 )
  • B ( 7 , 8 , 2 )
  • C ( 4 , βˆ’ 2 , βˆ’ 9 )
  • D ( βˆ’ 6 , βˆ’ 9 , βˆ’ 8 )
  • E ( 1 4 , 1 3 , βˆ’ 8 )

Q5:

Find the point of intersection of the planes βˆ’ 9 π‘₯ βˆ’ 8 𝑦 βˆ’ 2 𝑧 + 9 4 = 0 , βˆ’ 4 π‘₯ + 9 𝑦 βˆ’ 9 𝑧 βˆ’ 4 9 = 0 , and βˆ’ 5 π‘₯ + 8 𝑦 + 2 𝑧 + 1 8 = 0 .

  • A ( βˆ’ 4 , 9 , βˆ’ 9 )
  • B ( βˆ’ 9 , βˆ’ 8 , βˆ’ 2 )
  • C ( βˆ’ 5 , 8 , 2 )
  • D ( 8 , 4 , βˆ’ 5 )
  • E ( βˆ’ 1 8 , 9 , βˆ’ 9 )

Q6:

Find the vector equation of the line of intersection between the two planes π‘₯ + 3 𝑦 + 2 𝑧 βˆ’ 6 = 0 and 2 π‘₯ βˆ’ 𝑦 + 𝑧 + 2 = 0 .

  • A r = ( 0 , 1 4 , 1 2 ) + 𝑑 ( 5 , 3 , βˆ’ 7 )
  • B r = ( 0 , 2 , 0 ) + 𝑑 ( 2 , βˆ’ 3 , 2 )
  • C r = ( 0 , 1 4 , 1 2 ) + 𝑑 ( 2 , βˆ’ 3 , 2 )
  • D r = ( 0 , 2 , 0 ) + 𝑑 ( 5 , 3 , βˆ’ 7 )
  • E r = ( 0 , 2 , βˆ’ 1 2 ) + 𝑑 ( 5 , 3 , βˆ’ 7 )

Q7:

Find the vector equation of the line of intersection between the two planes 3 π‘₯ + 𝑦 βˆ’ 5 𝑧 = 0 and π‘₯ + 2 𝑦 + 𝑧 + 4 = 0 .

  • A ( 1 . 5 , 0 , 2 . 5 ) + 𝑑 ( 1 1 , βˆ’ 8 , 5 )
  • B ( 2 . 5 , 0 , 1 . 5 ) + 𝑑 ( 3 , 2 , βˆ’ 5 )
  • C ( 1 . 5 , 0 , 2 . 5 ) + 𝑑 ( 3 , 2 , βˆ’ 5 )
  • D ( βˆ’ 2 . 5 , 0 , βˆ’ 1 . 5 ) + 𝑑 ( 1 1 , βˆ’ 8 , 5 )
  • E ( 2 . 5 , 0 , 1 . 5 ) + 𝑑 ( 4 , 3 , βˆ’ 4 )

Q8:

𝐴 𝐡 𝐢 𝐴 β€² 𝐡 β€² 𝐢 β€² is an inclined prism, and 𝐡 𝐢 𝐢 β€² 𝐡 β€² is a square. Draw 𝐡 𝐷 βŸ‚ 𝐴 𝐴 β€² with 𝐷 on 𝐴 𝐴 β€² . Given that 𝐴 𝐡 = 1 9 , 𝐡 𝐷 = 8 , and 𝐴 𝐢 = 2 1 , find the length of 𝐷 𝐢 .

  • A29.4
  • B 3 √ 8 2
  • C20.6
  • D12