ABCD – parallelogram Vector AB = vector a, AD = vector b, K belongs to BC, L belongs to AD, BK: KC = 2: 3

ABCD – parallelogram Vector AB = vector a, AD = vector b, K belongs to BC, L belongs to AD, BK: KC = 2: 3, AL: LD = 3: 2 Find the decomposition of vector KL in non-collinear vectors a and b.

Since AВСD is a parallelogram, then AD = BC, AB = CD.

Then the vector ВС = АD = b.

By condition. ВC = 3/5 * BC, CК = 2/5 * BC, and the ВC vector = 3 * b / 5, the CК vector = 2 * b / 5.

Let’s construct a segment LH parallel to AB, then the vector HL = -AB = -a.

Segment BN = AL = 2 * b / 3.

Then НK = ВK – ВН = 3 * b / 5 – 2 * b / 5 = b / 5.

Vector KH = -b / 5.

Then the vector КL = KH + HL = – (b / 5) + (-a) = – (a + b / 5).

Answer: Vector KL = – (a + b / 5).