Lesson F3-F4

F3-F4 Scalar product and angles Quiz: AQA Further Maths, Unit 2

20 questions

Written by Revision Ninja · Last reviewed 11 October 2026

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Lesson F3-F4, Scalar product and angles: 20 multiple choice questions for the AQA Further Maths (7367), Unit 2: Compulsory content, written with Revision Ninja.

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The 20 questions

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

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

  2. What is the scalar product of (1, 2, 3) and (4, 5, 6)?

    • 15
    • 27
    • 32
    • 38

  3. If a and b are non-zero vectors and a . b = 0, what can be concluded?

    • |a| = |b|
    • a and b are parallel
    • a and b are perpendicular
    • a = b

  4. How is the angle between two planes found from their Cartesian equations?

    • From the distance of each plane from the origin
    • From the angle between their normal vectors
    • It is always 90 degrees
    • From the angle between their direction vectors of the intersection line

  5. The angle between a line and a plane is measured how?

    • It is always 90 degrees
    • As the complement of the angle between the line and the normal to the plane
    • As the angle between the line and the origin
    • As the angle between the line and the normal to the plane

  6. For any vector a, which result is always true?

    • a . a = 0
    • a . a = |a|
    • a . a = |a|^2
    • a . a = 2|a|

  7. What type of quantity is the scalar product of two vectors?

    • A scalar (a single number)
    • A vector of the same length as a
    • A matrix
    • A vector perpendicular to both

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

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

  9. Find the angle between the vectors (1, 2, 3) and (4, 5, 6), to 1 decimal place.

    • 60.0 degrees
    • 25.8 degrees
    • 77.1 degrees
    • 12.9 degrees

  10. Find the acute angle between lines with direction vectors (1, 2, 2) and (2, -1, 2), to 1 decimal place.

    • 26.4 degrees
    • 63.6 degrees
    • 90.0 degrees
    • 116.4 degrees

  11. What is the acute angle between the planes 2x + y - z = 3 and x - y + 2z = 1?

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

  12. The direction vector of a line is (1, 2, 2). What is the angle between this line and the plane z = 0?

    • arcsin(2/3)
    • arccos(1/3)
    • arccos(2/3)
    • arcsin(1/3)

  13. What is the angle between the line r = (1, 0, 0) + t(1, 1, 1) and the plane x + y + z = 5?

    • 90 degrees
    • 0 degrees
    • 45 degrees
    • arcsin(1/3)

  14. For the vector (1, lambda, 2) to be perpendicular to (2, -1, 1), what is lambda?

    • 4
    • 2
    • -4
    • 1

  15. For the vector (k, 1, 2) to make an angle of 60 degrees with (1, 0, 0), which value of k is correct?

    • k = -sqrt(15)/3
    • k = sqrt(15)/3
    • k = sqrt(5)
    • k = 5/3

  16. Find k so that the vectors (2, -1, 3) and (k, 4, 1) are perpendicular.

    • 1/2
    • -2
    • 2
    • -1/2

  17. Vectors a and b have |a| = 3, |b| = 2 and a . b = 3. What is the cosine of the angle between them?

    • 1/6
    • 1/2
    • 3
    • 3/2

  18. A line has direction vector (1, 0, 1) and the plane is 2x + y + z = 0. What is the angle between the line and the plane?

    • 60 degrees
    • 30 degrees
    • arccos(sqrt(3)/2)
    • 45 degrees

  19. Points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1) form a triangle. What is the size of angle ABC?

    • 60 degrees
    • 45 degrees
    • 120 degrees
    • 90 degrees

  20. If |a| = |b|, which vector is perpendicular to a + b?

    • a
    • 2a + b
    • a + 2b
    • a - b

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