6.3-6.4 The scalar product and checking perpendicularity Quiz: Pearson Edexcel Further Maths, Unit 6

The 20 questions

1. In component form, what is the scalar product a . b of a = (a1, a2, a3) and b = (b1, b2, b3)?

   • (a1 + b1)(a2 + b2)(a3 + b3)
   • a1 b1 - a2 b2 + a3 b3
   • a1 b2 + a2 b3 + a3 b1
   • a1 b1 + a2 b2 + a3 b3

2. Which expression gives the scalar product a . b in terms of the angle theta between the vectors?

   • |a| |b| tan theta
   • |a| |b| cos theta
   • |a| |b| sin theta
   • |a| + |b| cos theta

3. When are two non-zero vectors a and b perpendicular?

   • a . b = 1
   • |a| = |b|
   • a . b = 0
   • a x b = 0

4. What is the scalar product form of the equation of a plane with normal n and constant d?

   • r x n = d
   • r . n = d
   • n = r + d
   • r . n = 0 for every plane

5. What is the scalar product a . a in terms of |a|?

   • 2|a|
   • |a|
   • 0
   • |a|^2

6. What is (1, 2, 3) . (4, -5, 6)?

   • -12
   • 6
   • 14
   • 12

7. What is the angle between the vectors (1, 0, 0) and (1, 1, 0)?

   • 45 degrees (pi/4)
   • 60 degrees (pi/3)
   • 30 degrees (pi/6)
   • 90 degrees (pi/2)

8. Are the vectors (1, 2, 2) and (2, 1, -2) perpendicular?

   • No, their scalar product is -2
   • Yes, but only because their magnitudes are equal
   • Yes, their scalar product is zero
   • No, their scalar product is 2

9. What is the angle between the planes x + y = 1 and x + z = 2?

   • 30 degrees
   • 60 degrees
   • 90 degrees
   • 45 degrees

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

    • x + y + z = 1
    • x + 2y + 3z = 6
    • x + y + z = 3
    • x + y + z = 6

11. What is the magnitude of the vector (3, 4, 0)?

    • 7
    • 5
    • sqrt(7)
    • 25

12. What is (2, 3, -1) . (1, -1, 4)?

    • -3
    • -5
    • 5
    • 3

13. For the vectors (1, k, 2) and (3, 1, -1) to be perpendicular, what must k be?

    • k = 1
    • k = -1
    • k = -3
    • k = 2

14. What is the cosine of the angle between (1, 1, 1) and (1, 0, 0)?

    • sqrt(3)
    • 1/3
    • 1/sqrt(3)
    • 1

15. What is the angle between the lines with direction vectors (1, 0, 1) and (0, 1, 1)?

    • 30 degrees
    • 60 degrees
    • 45 degrees
    • 90 degrees

16. Is a line with direction (1, 2, 0) parallel to the plane 2x - y + 3z = 1?

    • Yes, since the line is perpendicular to the plane
    • Yes, since (1, 2, 0) . (2, -1, 3) = 0
    • No, since the scalar product is 2
    • No, since the scalar product is 4

17. What is the angle between the line r = lambda(1, 1, 1) and the plane x = 0?

    • 45 degrees
    • arcsin(sqrt(3))
    • arcsin(1/sqrt(3))
    • arccos(1/sqrt(3))

18. If a . b = 0, |a| = 3 and |b| = 4, what is |a + b|?

    • sqrt(7)
    • 1
    • 5
    • 7

19. For a = (1, 2, 3) and b = (2, k, 1) to be perpendicular, what is k?

    • -5/2
    • 2
    • 5/2
    • -2

20. The planes 2x + y - z = 3 and x - y + 2z = 1 have normals (2, 1, -1) and (1, -1, 2). What is the acute angle between the planes?

    • arccos(1/2)
    • arccos(-1/6)
    • arccos(1/3)
    • arccos(1/6)

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