The scalar product

Two formulae

a · b = a_1 b_1 + a_2 b_2 + a_3 b_3 (multiply matching components and add)

a · b = |a||b|cos θ, where θ is the angle between a and b.

Together: cos θ = (a · b)/(|a||b|).

Perpendicular vectors

If a and b are non-zero and a · b = 0, then cos θ = 0, so θ = 90°. Use this to find an unknown that makes two vectors perpendicular. Also a · a = |a|².

Angles in shapes

To find angle BAC in a triangle, use the vectors AB and AC, both pointing away from A. A negative scalar product means an obtuse angle.

Worked example

Find the angle between a = i + 2j + 2k and b = 3i + 4k.

1. a · b = 3 + 0 + 8 = 11
2. |a| = 3 and |b| = 5
3. cos θ = 11/15

Answer: θ = 42.8°

Check yourself

1. Find (3i - 2j + k) · (2i + 4j + 5k).
   Show answer

   3. 6 - 8 + 5 = 3.

2. Find p so that 2i + pj - k is perpendicular to 3i + j + 4k.
   Show answer

   -2. (2)(3) + p(1) + (-1)(4) = 0 gives 2 + p = 0, so p = -2.

3. Find the angle between i + j and j + k.
   Show answer

   60 °. a · b = 1, |a| = |b| = sqrt(2), so cos θ = 1/2 and θ = 60°.

4. Find the angle between 2i - j + 2k and i + 4j - 8k, to 1 decimal place.
   Show answer

   131.8 °. a · b = 2 - 4 - 16 = -18; |a| = 3 and |b| = 9. cos θ = -2/3, so θ = 131.8°.

5. The scalar product of two non-zero vectors is negative. What does this tell you?
   • The angle between them is more than 90°
   • They are perpendicular
   • They point in the same direction
   • One of them has negative length
   Show answer

   The angle between them is more than 90°. a · b = |a||b|cos θ and the lengths are positive, so cos θ < 0. That means 90° < θ ≤ 180°.

6. Which is equal to a · a?
   • |a|
   • 0
   • |a|²
   • 2|a|
   Show answer

   |a|². The angle between a and itself is 0, and cos 0 = 1, so a · a = |a|².

7. A(1, 0, 0), B(0, 2, 0) and C(0, 0, 3). Find angle BAC, to 1 decimal place.
   Show answer

   81.9 °. AB = (-1, 2, 0) and AC = (-1, 0, 3). AB · AC = 1, |AB| = sqrt(5), |AC| = sqrt(10). cos θ = 1/sqrt(50), so θ = 81.9°.

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Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 4 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.