F1-F2 Vector equations of lines and planes Quiz: AQA Further Maths, Unit 2

The 20 questions

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

   • The position vector of the origin
   • A normal to the line
   • A point on the line
   • A direction vector of the line

2. In the Cartesian equation ax + by + cz = d of a plane, which vector is normal to the plane?

   • (d, a, b)
   • (d, d, d)
   • (x, y, z)
   • (a, b, c)

3. If a plane has vector equation r = a + s b + t c, which vector is normal to the plane?

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

4. Which scalar product form gives the equation of a plane with normal n passing through a point with position vector a?

   • r x n = a x n
   • r . n = 0
   • r + n = a
   • r . n = a . n

5. Which point lies on the plane x + 2y - z = 2?

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

6. Find the Cartesian equation of the plane through the points (1, 0, 0), (0, 1, 0) and (0, 0, 1).

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

7. A line passes through (1, 2, 3) and (4, 0, -1). Which vector equation describes this line?

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

8. Find the Cartesian equation of the plane through (2, 1, -1) with normal vector (3, -1, 2).

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

9. Find the Cartesian equation of the plane through (1, 0, 0), (0, 2, 0) and (0, 0, 3).

   • 6x + 3y + 2z = 6
   • 6x + 3y + 2z = 0
   • 3x + 2y + z = 6
   • x + 2y + 3z = 6

10. The point (1, 2, k) lies on the plane 2x + 3y - z = 10. What is the value of k?

    • -6
    • -2
    • 18
    • 2

11. Write the line r = (1, 2, 3) + t(2, -1, 4) in Cartesian form.

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

12. Find the Cartesian equation of the plane through (1, 2, 3) with normal vector (2, -1, 1).

    • 2x + y + z = 3
    • 2x - y + z = 3
    • 2x - y - z = 3
    • 2x - y + z = 1

13. Find the direction vector of the line (x - 3)/2 = (y + 1)/(-4) = z/5.

    • (2, -4, 5)
    • (3, 1, 0)
    • (3, -1, 0)
    • (2, 4, 5)

14. What point is reached on the line r = (2, 0, 1) + t(1, 1, 0) when t = 3?

    • (5, 3, 0)
    • (3, 3, 0)
    • (5, 3, 1)
    • (2, 0, 1)

15. A plane has normal (1, -2, 2) and passes through the origin. Which point also lies on this plane?

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

16. The plane with vector equation r = (1, 0, 0) + s(1, 1, 0) + t(0, 1, 2) has which Cartesian form?

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

17. Find the Cartesian equation of the plane r = (1, 2, 0) + s(1, 0, 1) + t(0, 1, -1).

    • x + y + z = 3
    • -x + y + z = 3
    • -x + y + z = 1
    • x - y - z = 1

18. A line is given by x = 1 + 2t, y = 2 - t, z = 3 + 4t. What is the point on the line where x = 5?

    • (5, 0, 7)
    • (5, 1, 11)
    • (5, 0, 11)
    • (5, 2, 3)

19. The point (k, 1, 2) lies on the plane x + 2y - 3z = 4. What is k?

    • 4
    • 8
    • -8
    • 2

20. Find the vector equation of the line through the points P(2, -1, 3) and Q(5, 1, 0).

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

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