A vector in the plane: coordinates, length, dot product
Find the coordinates and length of a vector, the dot product by coordinates and by angle, the condition for perpendicular vectors and the angle between vectors.
- سوريا: الصف ١٠
- فلسطين: الصف ١١
- أوكرانيا: الصف ٩
- روسيا: الصف ٩
- ألمانيا: الصف ١٠
- فرنسا: الصف ٩
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الدليل
In Ukrainian Grade 9 geometry (section 4) a vector in the plane is given by coordinates: AB (x٢ − x١;y٢ − y١), where A (x١;y١) and B (x٢;y٢). The length of a (a١;a٢) is |a| = √(a١٢ + a٢٢).
The angle between two non-zero vectors a and b is the angle AOB, where OA = a, OB = b. It is written ∠(a, b); it is 0° for vectors in the same direction and 180° for opposite directions. Vectors are perpendicular when the angle is 90°: a ⊥ b.
The scalar (dot) product of two non-zero vectors is the product of their lengths and the cosine of the angle between them: a · b = |a| |b| cos ∠(a, b). Theorem (coordinates): a · b = a١b١ + a٢b٢. Vectors are perpendicular exactly when a · b = 0.
The cosine of the angle between non-zero vectors: cos∠(a,b) = a١b١ + a٢b٢√(a١٢ + a٢٢)·√(b١٢ + b٢٢).