On the sides BC and CD of the parallelogram ABCD, points K and E are marked so that BK = KC, CE: ED = 2: 3. Express the vectors AK, AE, KE through the vectors a = AB and u = AD
Vector AK = AB + BK = AB + BC / 2 = a + b / 2.
Since ABCD is a parallelogram, then CD = AB = a. Then 2 * ED = 3 * CE. EC = 2 * ED / 3.
CD = a = ED + 2 * ED / 3 = 5 * ED / 3.
EB = 3 * a / 5.
Vector AE = AD + DE = b + 3 * a / 5.
Vector KE = AE – AK = b + 3 * a / 5 – a – b / 2 = b / 2 – 2 * a / 5.
Answer: AK = a + b / 2, AE = b + 3 * a / 5, KE = b / 2 – 2 * a / 5.
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