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.
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Higher Mathematics
Scalar product
Use the scalar product to find angles and test perpendicular 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
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 = a · b|a||b|.
So: cos θ = 1/√2, 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.