6.5 Perpendicular distance from a point to a line or plane Quiz: Pearson Edexcel Further Maths, Unit 6

The 20 questions

1. What is the perpendicular distance from the point (alpha, beta, gamma) to the plane n1 x + n2 y + n3 z + d = 0?

   • |n1 alpha + n2 beta + n3 gamma + d| x sqrt(n1^2 + n2^2 + n3^2)
   • (n1 alpha + n2 beta + n3 gamma + d) / (n1^2 + n2^2 + n3^2)
   • |n1 alpha + n2 beta + n3 gamma + d| / sqrt(n1^2 + n2^2 + n3^2)
   • |n1 alpha + n2 beta + n3 gamma + d| / (n1 + n2 + n3)

2. In finding the perpendicular from a point to a plane, along which direction does the perpendicular line travel?

   • Parallel to the normal vector of the plane
   • Along the x-axis
   • Parallel to the plane itself
   • Perpendicular to the normal vector of the plane

3. What is the distance between the parallel planes n . r = d1 and n . r = d2?

   • |d1 - d2| x |n|
   • |d1 + d2| / |n|
   • |d1 - d2|
   • |d1 - d2| / |n|

4. Which method finds the perpendicular distance from a point to a line in 3-D?

   • Use the plane distance formula with the line direction as the normal
   • Find the point where the line meets the x-axis and use its distance
   • Divide the line direction by the point position vector
   • Take a general point on the line, make the connecting vector perpendicular to the line, then find its length

5. What is the perpendicular distance from the point (3, 1, -2) to the plane x + 2y + 2z = 9?

   • 8/3
   • 8
   • 4
   • 8/9

6. What is the distance from the origin to the plane 3x + 4y = 10?

   • 10
   • 2
   • 10/7
   • 5/2

7. What is the perpendicular distance from the point (1, 2, 3) to the plane x + y + z = 0?

   • 2/sqrt(3)
   • 6
   • 2 sqrt(3)
   • sqrt(6)

8. What is the foot of the perpendicular from the origin to the plane x + y + z = 3?

   • (1, 1, 1)
   • (1, 1, 0)
   • (3/2, 3/2, 0)
   • (3, 0, 0)

9. What is the distance between the parallel planes x + 2y + 2z = 3 and x + 2y + 2z = 9?

   • 2
   • 2/3
   • 3
   • 6

10. Where does the line r = (1, 0, 0) + lambda(0, 1, 1) meet the plane x + y = 4?

    • (1, 2, 2)
    • (1, 4, 4)
    • (1, 3, 3)
    • (4, 0, 0)

11. What is the perpendicular distance from the point (2, 3, 4) to the line r = lambda(0, 1, 0)?

    • 2 sqrt(5)
    • sqrt(29)
    • 7
    • 5

12. What is the perpendicular distance from the point (0, 0, 5) to the plane z = 2?

    • 3
    • 5
    • 7
    • 2

13. What is the perpendicular distance from the point (1, 1, 1) to the plane 2x + 2y + z = 1?

    • 1/3
    • 5/3
    • 4
    • 4/3

14. What is the perpendicular distance from the point (2, -1, 1) to the plane x - 2y + 2z + 4 = 0?

    • 10/9
    • 10
    • 2
    • 10/3

15. What is the perpendicular distance from the point (1, 2, 3) to the line r = lambda(1, 0, 0)?

    • 5
    • sqrt(5)
    • sqrt(13)
    • 3

16. What is the perpendicular distance from the point (1, 2, 3) to the line r = (0, 1, 0) + lambda(1, 1, 1)?

    • 2 sqrt(6)/3
    • sqrt(6)
    • 4/3
    • 2 sqrt(3)/3

17. What is the foot of the perpendicular from the point (1, 2, 3) to the plane x + y + z = 0?

    • (-1, 0, -1)
    • (-1, 0, 1)
    • (1, 2, 3)
    • (0, 1, 2)

18. What is the perpendicular distance from the point (3, 1, 1) to the plane through the origin with normal (0, 1, 1)?

    • 2 sqrt(2)
    • 1
    • 2
    • sqrt(2)

19. What is the distance between the parallel planes x + 2y + 2z = 3 and 2x + 4y + 4z = 15?

    • 1/2
    • 3
    • 9/2
    • 3/2

20. What is the shortest distance between the skew lines r = lambda(1, 0, 0) and r = (0, 0, 1) + mu(0, 1, 0)?

    • 1
    • sqrt(2)
    • 2
    • 0

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