The dot product of vectors: lengths, angle, coordinates
Find the dot product from lengths and the angle, square a sum of vectors, find the cosine of the angle between vectors and the angle of a triangle from its vertices.
- France: grade 11
- Germany: grade 12
- Syria: grade 11
- Palestine: grade 11
- Ukraine: grade 10
- Russia: grade 9
Guide
In Ukrainian Grade 10 (geometry, section 6, topic 42) the angle between non-zero vectors a and b is the angle AOB, where OA = a, OB = b. It is written ∠(a, b), 0° ≤ ∠(a,b) ≤ 180°. Vectors are perpendicular if the angle is 90°.
The scalar (dot) product is the product of the lengths and the cosine of the angle between them: a · b = |a| |b| cos ∠(a, b). The scalar square a · a is written a², and a² = |a|². Vectors are perpendicular exactly when a · b = 0.
For a (a1;a2;a3) and b (b1;b2;b3): a·b = a1b1 + a2b2 + a3b3. Hence cos∠(a,b) = a1b1 + a2b2 + a3b3√(a12 + a22 + a32)·√(b12 + b22 + b32).
The dot product is commutative, (ka) · b = k(a · b) and distributive over addition. So (a + b)² = a² + 2a · b + b².