Worked example 1
Calculate a dot product
a = (3, −2), b = (4, 5). Find a · b.
- Multiply matching components.
- Add the products.
Answer: a · b = 3×4 + (-2)×5 = 2.
Higher Mathematics
Higher Maths Scalar product
Practise scalar product for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Calculate scalar products, test perpendicularity and find the angle between vectors.
Before you start
- Find vector magnitudes.
- Multiply and add components accurately.
- Rearrange cos θ formulae.
Explanation
The scalar product, or dot product, combines two vectors to give a number. It helps find angles between vectors and test for perpendicular vectors.
If a · b = 0 and neither vector is zero, the vectors are perpendicular.
Visual support
a · b = |a||b| cos θ = a₁b₁ + a₂b₂
Method and rules
- a · b = a₁b₁ + a₂b₂.
- a · b = |a||b|cos θ.
- If a · b = 0, then a and b are perpendicular.
Worked examples
Worked example 2
Find an angle
For a = (1, 0), b = (1, 1), find the angle between them.
- Calculate a · b = 1
- Find |a| = 1 and |b| = √2.
- Use cos θ = .
So: cos θ = , so θ = 45°
Watch out
- Adding components instead of multiplying matching components.
- Forgetting the square root when finding magnitudes.
- Rearranging the angle formula upside down.
- Rounding the angle too early.
Exam reminder
State whether the angle is exact or rounded. If using a calculator, give the degree of accuracy requested.
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