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(11,3,10)

Q2:

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

  • Ar=(0,2,12)+𝑡(5,3,7)
  • Br=(0,14,12)+𝑡(2,3,2)
  • Cr=(0,2,0)+𝑡(2,3,2)
  • Dr=(0,2,0)+𝑡(5,3,7)
  • Er=(0,14,12)+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(11,8,5)
  • C(1.5,0,2.5)+𝑡(3,2,5)
  • D(2.5,0,1.5)+𝑡(11,8,5)
  • 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(14,9,10)

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(14,13,8)

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(18,9,9)

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(6,8,14)
  • C(7,9,13)
  • 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?

  • A92
  • B211
  • C112
  • D29

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(18,0,3)
  • B(10,2,1)
  • C(2,4,1)
  • D(2,4,1)
  • E(24,3,0)

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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