Worksheet: Intersection of Planes

In this worksheet, we will practice finding the point/line of intersection between planes given their equations.

Q1:

Find the point of intersection of the planes 5𝑥2𝑦+6𝑧1=0, 7𝑥+8𝑦+𝑧6=0, and 𝑥3𝑦+3𝑧+11=0.

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

Q2:

Find the vector equation of the line of intersection between the two planes 𝑥+3𝑦+2𝑧6=0 and 2𝑥𝑦+𝑧+2=0.

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

Q3:

Find the vector equation of the line of intersection between the two planes 3𝑥+𝑦5𝑧=0 and 𝑥+2𝑦+𝑧+4=0.

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

Q4:

Find the point of intersection of the planes 8𝑥+5𝑦𝑧17=0, 2𝑥4𝑦7𝑧+85=0, and 6𝑥+𝑦+3𝑧41=0.

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

Q5:

Find the point of intersection of the planes 5𝑥3𝑦+7𝑧+5=0, 7𝑥+3𝑦+𝑧5=0, and 2𝑥9𝑦+2𝑧16=0.

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

Q6:

Find the point of intersection of the planes 7𝑥+8𝑦+2𝑧+130=0, 3𝑥+7𝑦𝑧+73=0, and 4𝑥2𝑦9𝑧66=0.

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

Q7:

Find the point of intersection of the planes 9𝑥8𝑦2𝑧+94=0, 4𝑥+9𝑦9𝑧49=0, and 5𝑥+8𝑦+2𝑧+18=0.

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

Q8:

Find the coordinates of the point of intersection of the straight line r=8,2,5+𝑡7,9,13 with the plane 9,4,5=59r.

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

Q9:

Straight line 𝐴𝐵 is parallel to plane 𝑋, and from a point 𝑀 neither on the line nor in the plane are drawn rays 𝑀𝐴, 𝑀𝐵 meeting 𝑋 in 𝐷 and 𝐻. If 𝑀𝐴𝐴𝐷=29::, what is the ratio between 𝐴𝐵 and 𝐷𝐻?

  • A 9 2
  • B 2 1 1
  • C 1 1 2
  • D 2 9

Q10:

Find the point of intersection of the straight line 3𝑥=4𝑦2=𝑧+1 and the plane 3𝑥+𝑦+𝑧=13.

  • A ( 2 , 2 , 5 )
  • B ( 2 , 2 , 5 )
  • C ( 5 , 2 , 2 )
  • D ( 2 , 5 , 2 )

Q11:

Find the point of intersection of the line 𝑥64=𝑦+3=𝑧 with the plane 𝑥+3𝑦+2𝑧6=0.

  • A ( 1 8 , 0 , 3 )
  • B ( 1 0 , 2 , 1 )
  • C ( 2 , 4 , 1 )
  • D ( 2 , 4 , 1 )
  • E ( 2 4 , 3 , 0 )

Q12:

Determine whether the following sentence is true or false: In every plane, any two intersecting lines are perpendicular.

  • Atrue
  • Bfalse

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