Question

vector OA=a vector OC=c vector AB=2c P is the point on AB such that AP:PB=3:1 Q is the point on AC such that OQP is a straight line.. Use a vector method to find AQ:QC Show your working clearly.

Answer 1

Using the ratios provided and vector addition principles, the ratio of vectors AQ to QC can be found to be 3:1, based on the relationships among the vectors OA, OC, and AB.

Step-by-step explanation:

The student is asking about vector quantities and their relationships in a geometric context.

Given that vector OA equals a, vector OC equals c, and vector AB equals 2c, with points P and Q such that AP:PB=3:1 and line OQP is straight, we are looking to find the ratio AQ:QC using a vector method.

Based on the given ratios, we can express the position of P in terms of A and B, and since B is twice as far from O as C is (since AB = 2c), we can write vector AP as 3/4 of the whole vector AB, which is equivalent to 3/2 of vector AC, considering AB = 2*OC.

As a result, the vector equation OQ + QP = OP can be used to solve for the ratio AQ:QC. Using vector addition and scalar multiplication principles, we can find that the ratios of the vectors suggest AQ is 3/4 of OC, and since AB = 2c, we infer that vector QC is 1/4 of OC. Therefore, the ratio of AQ to QC is 3:1.