Worksheet: Angle between Two Planes

In this worksheet, we will practice finding the measure of an angle between two planes or that between a line and a plane.

Q1:

Find, to the nearest second, the measure of the angle between the planes 9𝑥6𝑦+5𝑧=8 and 2𝑥+2𝑦+7𝑧=8.

  • A 4 8 3 1 3 7
  • B 6 5 1 1 2 9
  • C 8 6 4 8 5 1
  • D 7 5 6 5 9

Q2:

Find, to the nearest second, the measure of the angle between the planes 5,9,4=6r and 4𝑥𝑦6𝑧=3.

  • A 6 3 5 4
  • B 4 6 3 7 2 1
  • C 7 4 9 5 9
  • D 8 6 2 6 6

Q3:

Find, to the nearest second, the measure of the angle between the straight line rijkijk=(9+85)+𝑡(+3) and the normal to the plane rijk(3+2+5)=7.

  • A 5 1 2 9 5 3
  • B 1 2 8 3 0 7
  • C 2 6 3 6 3 4
  • D 3 8 3 0 7

Q4:

Which of the following is the smaller angle between the straight line rijkijk=(79)+𝑡(2+) and the plane rijk(99+2)=13?

  • A s i n 7 4 3 8 6
  • B s i n 7 2 4 9 4 9 8
  • C c o s 7 2 4 9 4 9 8
  • D c o s 7 4 3 8 6

Q5:

Find, to the nearest second, the measure of the smaller angle between the straight line 𝑥77=𝑦75=𝑧41 and the plane 6𝑥8𝑦5𝑧17=0.

  • A 5 2 4 0 4 5
  • B 1 6 3 2 5 3
  • C 1 2 7 1 9 1 5
  • D 3 7 1 9 1 5

Q6:

If the angle between the two planes 𝑎𝑥4𝑦+5𝑧=5 and r(7,1,0)=4 is 60, find the value of the positive constant 𝑎.

Q7:

Find, to the nearest second, the measure of the angle between the straight line 𝑥=3𝑡1,𝑦=2𝑡+4,𝑧=5 and the plane 3𝑥4𝑦+𝑧=2.

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

Q8:

Find, to the nearest degree, the measure of the angle between the planes 2(𝑥1)+3(𝑦4)+4(𝑧+5)=0 and r(1,2,5)=16.

Q9:

Find, to the nearest degree, the measure of the angle between the planes 3𝑥2𝑦+3𝑧=7 and 2(𝑥1)+3(𝑦4)+4(𝑧+5)=0.

Q10:

If the angle between the plane 3𝑥+𝑏𝑦5𝑧=2 and the straight line 𝑥+2=3𝑦2=𝑧12 is 45, find the value of the positive constant 𝑏 given that 0<𝑏<10.

Q11:

Find, to the nearest degree, the measure of the angle between the plane 2(𝑥3)8(𝑦+2)+3𝑧=4 and the straight line𝑥36=𝑦+12=𝑧14.

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