Worksheet: Relationship between a Straight Line and a Plane

In this worksheet, we will practice determining the relationship between a straight line and a plane.

Q1:

The equation of a straight line is 𝐿 𝑥 1 2 = 𝑦 + 9 7 = 𝑧 + 5 5 and the equaton of a plane is 𝑃 8 𝑥 2 8 𝑦 2 0 𝑧 + 1 9 = 0 . Which of the following describes the relationship between 𝐿 and 𝑃 ?

  • A 𝐿 𝑃
  • B 𝐿 𝑃
  • CThey meet at a point but are not perpendicular.
  • D 𝐿 𝑃

Q2:

Find the coordinates of the point of intersection of the straight line 𝑟 = ( 8 , 2 , 5 ) + 𝑡 ( 7 , 9 , 1 3 ) with the plane ( 9 , 4 , 5 ) 𝑟 = 5 9 .

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

Q3:

Which of the following makes the straight line 𝑥 𝑥 𝑙 = 𝑦 𝑦 𝑚 = 𝑧 𝑧 𝑛 1 1 1 lie on the plane 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 + 𝑑 = 0 ?

  • A 𝑎 𝑙 + 𝑏 𝑚 + 𝑐 𝑛 = 0
  • B 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 = 0 1 1 1
  • C ( 𝑙 , 𝑚 , 𝑛 ) × ( 𝑎 , 𝑏 , 𝑐 ) = 0
  • D 𝑎 𝑙 + 𝑏 𝑚 + 𝑐 𝑛 = 0 and 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑧 = 0 1 1 1
  • E ( 𝑙 , 𝑚 , 𝑛 ) ( 𝑎 , 𝑏 , 𝑐 ) 0

Q4:

In the figure, 𝐴 𝐵 lies in the plane 𝑋 , and 𝐴 𝐶 is perpendicular to 𝑋 . Given that 𝐴 𝐵 = 6 and 𝐴 𝐶 = 8 , find the length of 𝐵 𝐶 .

Q5:

𝐴 𝐵 𝐶 is a triangle with 𝑚 𝐵 = 6 0 and 𝐵 𝐶 = 2 3 . 𝐶 𝐷 is drawn perpendicular to the plane of 𝐴 𝐵 𝐶 , and the perpendicular to 𝐴 𝐵 from 𝐷 drawn to meet it at 𝐸 . If 𝐷 𝐸 = 2 3 , determine the length of 𝐶 𝐷 and the angle between 𝐵 𝐷 and the plane of 𝐶 𝐷 𝐸 .

  • A19.92, 4 0 5 3 3 6 . 2 2
  • B30.43, 3 0
  • C19.92, 6 3 2 6 5 . 8 2
  • D11.5, 2 6 3 3 5 4 . 1 8
  • E11.5, 6 3 2 6 5 . 8 2

Q6:

Straight line 𝐴 𝐵 is parallel to plane 𝑋 , and from a point 𝑀 neither on the line nor in the plane are drawn rays 𝑀 𝐴 , 𝑀 𝐵 meeting 𝑋 in 𝐷 and 𝐻 . If 𝑀 𝐴 𝐴 𝐷 = 2 9 : : , what is the ratio between 𝐴 𝐵 and 𝐷 𝐻 ?

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

Q7:

What is 𝐴 𝐶 𝐴 𝐶 ?

  • A 𝐴 𝐴
  • B 𝐶 𝐶
  • C 𝐴 𝐶
  • D

Q8:

Triangle is right angled at , and is orthogonal to the plane . A perpendicular is drawn from on . The area of is 1 134, , and . Let be the angle between and the plane . Find to the nearest thousandth.

Q9:

In which of the following cases is the straight line 𝐴 𝐵 parallel to the plane 𝑋 ?

  • A 𝐴 𝐵 𝑋 = 𝐴 𝐵
  • B 𝐴 𝐵 𝑋 = { 𝐵 }
  • C 𝐴 𝐵 𝑋 = { 𝐴 }
  • D 𝐴 𝐵 𝑋 =
  • E 𝐴 𝐵 𝑋

Q10:

Find the point of intersection of the line 𝑥 6 4 = 𝑦 + 3 = 𝑧 with the plane 𝑥 + 3 𝑦 + 2 𝑧 6 = 0 .

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

Q11:

Find, to the nearest second, the measure of the smaller angle between the straight line 𝑥 7 7 = 𝑦 7 5 = 𝑧 4 1 and the plane 6 𝑥 8 𝑦 5 𝑧 1 7 = 0 .

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

Q12:

Which of the following scenarios would make 𝐴 𝐵 plane 𝑋 ?

  • A The distance from 𝐴 to 𝑋 is not the same as the distance from 𝐵 to 𝑋 .
  • B 𝐴 and 𝐵 lie on two different sides of 𝑋 .
  • C 𝐴 𝐵 𝑋 = .
  • D 𝐴 𝐵 𝑋 = .

Q13:

The given figure shows a pyramid placed on plane 𝑋 . Find the intersection of the plane through points 𝐻 𝐺 𝐷 and plane 𝑋 ?

  • A 𝐷 𝐻
  • B 𝐺 𝐷
  • C 𝐽 𝐻
  • D 𝐻 𝐺
  • E 𝐺 𝐹

Q14:

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

  • A false
  • B true

Q15:

Express the normal form of the plane containing the lines 𝑥 = 7 + 3 𝑠 , 𝑦 = 4 3 𝑠 , 𝑧 = 7 5 𝑠 and 𝑥 = 1 + 6 𝑡 , 𝑦 = 2 + 𝑡 , 𝑧 = 3 2 𝑡 .

  • A 1 1 𝑥 2 4 𝑦 + 2 1 𝑧 + 2 6 = 0
  • B 1 8 𝑥 3 𝑦 + 1 0 𝑧 6 8 = 0
  • C 1 8 𝑥 3 𝑦 + 1 0 𝑧 + 6 8 = 0
  • D 1 1 𝑥 2 4 𝑦 + 2 1 𝑧 2 6 = 0
  • E 9 𝑥 2 𝑦 3 𝑧 9 2 = 0

Q16:

Let 𝑆 be the surface you get by removing the point ( 0 , 0 , 2 ) from the sphere with centre ( 0 , 0 , 1 ) and radius 1, and let ( 𝑎 , 𝑏 , 𝑐 ) be an arbitrary point on 𝑆 . The line through ( 0 , 0 , 2 ) and ( 𝑎 , 𝑏 , 𝑐 ) intersects the 𝑥 𝑦 -plane at ( 𝑥 , 𝑦 , 0 ) , as shown in the figure. Find the coordinates of this point in terms of 𝑎 , 𝑏 , and 𝑐 .

  • A 2 𝑎 1