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.
- France: grade 9
- Germany: grade 10
- Syria: grade 10
- Palestine: grade 11
- Ukraine: grade 9
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Guide
In Ukrainian Grade 9 geometry (section 4) a vector in the plane is given by coordinates: AB (x2 − x1;y2 − y1), where A (x1;y1) and B (x2;y2). The length of a (a1;a2) is |a| = √(a12 + a22).
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 = a1b1 + a2b2. Vectors are perpendicular exactly when a · b = 0.
The cosine of the angle between non-zero vectors: cos∠(a,b) = a1b1 + a2b2√(a12 + a22)·√(b12 + b22).