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

   • A direction vector parallel to the line
   • The point of intersection with another line
   • A normal vector perpendicular to the line
   • The position vector of the origin

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 Cartesian form of a plane?

   • a . x = 0 for the origin only
   • x = y = z
   • ax + by = d
   • ax + by + cz = d

5. When are two lines in 3-D described as skew?

   • They intersect at right angles
   • They are parallel
   • They are not parallel and do not intersect
   • They are coincident

6. When are two lines in 3-D parallel?

   • Their direction vectors are perpendicular
   • Their direction vectors are scalar multiples of each other
   • They share a common point
   • Their direction vectors have the same length

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 in the plane x - y = 1?

    • No, it is parallel to the plane but not contained in it
    • No, it meets the plane only at (1, 0, 0)
    • Yes, but only when lambda = 0
    • Yes, every point on the line satisfies x - y = 1

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

    • Intersecting at the origin
    • 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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