The dot product of vectors whose components are known can be calculated with a simple formula.
Dot Product by Components
For →a=(a₁, a₂) and →b=(b₁, b₂), →a・→b = a₁b₁ + a₂b₂.
Example ①
When →a=(2, −3) and →b=(4, 1), find →a・→b.
Solution: 2×4 + (−3)×1 = 8 − 3 = 5
When →a=(2, −3) and →b=(4, 1), find →a・→b.
Solution: 2×4 + (−3)×1 = 8 − 3 = 5
Cosine of the Included Angle
cos θ = (a₁b₁ + a₂b₂) / (|→a| × |→b|)
Example ②
Find the angle between →a=(1, √3) and →b=(√3, 1).
Solution: →a・→b = 2√3, |→a|=|→b|=2. cos θ = √3/2 → θ=30°
Find the angle between →a=(1, √3) and →b=(√3, 1).
Solution: →a・→b = 2√3, |→a|=|→b|=2. cos θ = √3/2 → θ=30°
Key Points
- Dot product of components: the sum of the products of corresponding components
- Combine with magnitude √(x²+y²) to find cos θ
Practice Problems
- When →a=(3, 4) and →b=(−1, 2), find →a・→b.
- For the angle θ between →a=(1, 2) and →b=(3, −1), find cos θ.
- For →a=(2, 4) and →b=(3, 1), find cos θ.
Answers and Explanations
- Answer: 5
Explanation: 3×(−1)+4×2=5. - Answer: cos θ = √2/10
Explanation: Dot product 1, |→a|=√5, |→b|=√10, so cos θ=1/√50=√2/10. - Answer: cos θ=√2/2
Explanation: Dot product 10, |→a|=2√5, |→b|=√10. 10/(2√50)=10/(10√2)=√2/2.