Among the “multiplications” of vectors, the product whose result is a scalar (real number) is called the dot product (scalar product).
Definition of the Dot Product
Let θ（0° ≦ θ ≦ 180°）be the angle between two nonzero vectors →a and →b. Then,
→a・→b = |→a||→b|cos θ
Geometric Meaning
- θ=0°（same direction）→ →a・→b = |→a||→b|
- θ=90°（perpendicular）→ →a・→b = 0
- θ=180°（opposite directions）→ →a・→b = −|→a||→b|
- Acute angle → the dot product is positive; obtuse angle → the dot product is negative
Example ①
When |→a|=3, |→b|=4, and the angle between them is 60°, find →a・→b.
Solution: 3×4×cos60° = 12×(1/2) = 6
When |→a|=3, |→b|=4, and the angle between them is 60°, find →a・→b.
Solution: 3×4×cos60° = 12×(1/2) = 6
Properties of the Dot Product
- →a・→a = |→a|²
- →a・→b = →b・→a（commutative law）
- (k→a)・→b = k(→a・→b)
- →a・(→b + →c) = →a・→b + →a・→c（distributive law）
Example ②
When |→a|=2, |→b|=√3, and →a・→b=−3, find (→a + →b)・(2→a − →b).
Solution: 2|→a|² + →a・→b − |→b|² = 8 + (−3) − 3 = 2
When |→a|=2, |→b|=√3, and →a・→b=−3, find (→a + →b)・(2→a − →b).
Solution: 2|→a|² + →a・→b − |→b|² = 8 + (−3) − 3 = 2
Key Points
- The result of a dot product is a scalar (real number)
- →a・→b = |→a||→b|cos θ, →a・→a = |→a|²
- The distributive law and scalar multiplication hold, so expressions can be expanded like polynomials
Practice Problems
- When |→a|=4, |→b|=3, and the angle between them is 120°, find →a・→b.
- When |→a|=5, find →a・→a.
- When |→a|=2, |→b|=3, and →a・→b=4, find |→a + →b|².
Answers and Explanations
- Answer: −6
Explanation: 4×3×cos120° = 12×(−1/2) = −6. - Answer: 25
Explanation: →a・→a = |→a|² = 25. - Answer: 21
Explanation: |→a+→b|²=4+8+9=21.