ABCD – parallelogram, O – intersection point of diagonals, M – midpoint of AC, AB = a, AD = b. Express the following vectors in terms of vectors a and b: 1) AC; 2) AO; 3) BD; 4) AM; 5) DM.
The parallelogram property says: at the point of intersection, the diagonals are halved. Hence the points O and M coincide.
1) AC = AB + AD = a + b;
2) AO = 1/2 * AC = 1/2 (a + b);
3) BD = AB – AD = a – b;
4) AM = AO = 1/2 (a + b);
5) DM = 1/2 * DB = 1/2 (a – b).
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