ABCD – parallelogram, O – intersection point of diagonals, M – midpoint of BC, AB = a

ABCD – parallelogram, O – intersection point of diagonals, M – midpoint of BC, AB = a, AD = b. Express the following vectors in terms of vectors a and b: 1) AC; 2) AO; 3) BD;

Vector AD = BC, then the AC vector is equal to the sum of the AB and BC vectors.

Vector AC = AB + BC = a + b.

The diagonals of the parallelepiped, at the point of intersection, are divided in half, then AO = AC / 2.

Then the vector AO = AC / 2.

Vector AO = (a + b) / 2.

Vector AD = AB + BD, then vector BD = AD – AB = b – a.

The vector BM is equal to BC / 2, then the vector AM = AB + BC / 2 = a + (b / 2).

Answer: AC = a + b; AO = (a + b) / 2; BD = b – a; AM = a + (b / 2).