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.
- سوريا: الصف ١١
- فلسطين: الصف ١١
- أوكرانيا: الصف ١٠
- روسيا: الصف ٩
- ألمانيا: الصف ١٢
- فرنسا: الصف ١١
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الدليل
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), ٠° ≤ ∠(a,b) ≤ ١٨٠°. 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 (a١;a٢;a٣) and b (b١;b٢;b٣): a·b = a١b١ + a٢b٢ + a٣b٣. Hence cos∠(a,b) = a١b١ + a٢b٢ + a٣b٣√(a١٢ + a٢٢ + a٣٢)·√(b١٢ + b٢٢ + b٣٢).
The dot product is commutative, (ka) · b = k(a · b) and distributive over addition. So (a + b)² = a² + 2a · b + b².