6.1-6.2 Vector equations of lines and planes Quiz: Pearson Edexcel Further Maths, Unit 6

The 20 questions

1. What is a vector equation of a straight line in 3-D?

   • r = a + lambda b
   • r = a + b
   • r = a.b + lambda
   • r . n = d

2. In the vector line equation r = a + lambda b, what does vector b represent?

   • Direction vector
   • Scalar distance
   • Normal vector
   • Position vector

3. What is the vector form of a plane with a point a and two non-parallel direction vectors b and c?

   • r . b = 0 only
   • r = a + lambda b
   • r = lambda b + mu c + (a . c)
   • r = a + lambda b + mu c

4. What is the standard Cartesian equation of a plane in 3D space?

   • ax + by = c
   • y = mx + c
   • x^2 + y^2 + z^2 = r^2
   • ax + by + cz = d

5. What term describes two non-parallel, non-intersecting lines in 3D space?

   • Perpendicular
   • Coincident
   • Skew
   • Parallel

6. Two lines in 3D space are parallel if their direction vectors are what?

   • Unit vectors
   • Scalar multiples
   • Perpendicular
   • Orthogonal

7. For a plane given by r = a + lambda b + mu c, which vector is normal to the plane?

   • b . c
   • b x c
   • a x b
   • b + c

8. What is the Cartesian form of the line through (1, 2, 3) with direction (2, -1, 4)?

   • (x - 1)/2 = (y - 2)/1 = (z - 3)/4
   • (x - 1)/2 = (y - 2)/(-1) = (z - 3)/4
   • (x - 1)/4 = (y - 2)/(-1) = (z - 3)/2
   • (x - 2)/2 = (y - 1)/(-1) = (z - 3)/4

9. What point lies on the line r = (1, 2, 3) + lambda(2, -1, 4) when lambda = 1?

   • (2, 1, 4)
   • (1, 1, 7)
   • (3, 1, 7)
   • (3, 2, 7)

10. Which plane passes through the points (1, 0, 0), (0, 1, 0) and (0, 0, 1)?

    • x + y + z = 0
    • x + y + z = 1
    • x + y - z = 1
    • x - y - z = 1

11. Find the point of intersection of the lines r = lambda(1, 1, 0) and r = (2, 0, 0) + mu(-1, 1, 0).

    • (1, 1, 0)
    • (0, 1, 0)
    • (2, 0, 0)
    • (1, 0, 1)

12. Does the line r = (1, 0, 0) + lambda(1, 1, 1) lie entirely in the plane x - y = 1?

    • No, it intersects once
    • No, it is parallel
    • No, it is perpendicular
    • Yes

13. Are the lines r = lambda(1, 0, 0) and r = (0, 1, 1) + mu(0, 1, 0) skew, intersecting or parallel?

    • Intersecting
    • Coincident
    • Skew
    • Parallel

14. Find the vector equation of the line through (4, 0, -1) parallel to (3, 2, -2).

    • r = (1, 2, 3) + lambda(3, 2, -2)
    • r = (4, 0, -1) + lambda(3, -2, 2)
    • r = (3, 2, -2) + lambda(4, 0, -1)
    • r = (4, 0, -1) + lambda(3, 2, -2)

15. What is the Cartesian equation of the plane through (1, 2, 3) with normal (2, -1, 3)?

    • 2x + y + 3z = 9
    • 2x - y + 3z = 6
    • 2x - y + 3z = 3
    • 2x - y + 3z = 9

16. What is the Cartesian equation of the plane r = (1, 0, 2) + lambda(1, 1, 0) + mu(0, 0, 1)?

    • x - y = 0
    • x - y = 2
    • x - y = 1
    • x + y = 1

17. Find the point of intersection of the lines r = (1, 0, 0) + lambda(1, 2, 0) and r = (0, 4, 0) + mu(1, -1, 0).

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

18. A line r = a + lambda b lies in the plane n . r = d. Which conditions must hold?

    • b . n = 0 and a . n = d
    • a . n = 0 only
    • b . n = d and a . n = 0
    • b . n != 0 and a . n = d

19. What is the Cartesian equation of the plane r = lambda(1, 2, 0) + mu(0, 1, 1)?

    • 2x + y - z = 0
    • x - 2y + z = 0
    • 2x - y + z = 0
    • 2x - y + z = 1

20. Which Cartesian equation describes the plane through (1, 0, 0) with directions (0, 1, 0) and (0, 0, 1)?

    • x + y + z = 1
    • y = 1
    • x = 1
    • z = 0

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