02) 2D Vector: Ratios



In this scenario, Point P lies on line AB, and ratio AP : PB is 1 : 3. However what does this mean?

First, the ratio implies that PB is 3 times the length of AP,.
Second, As Points A, P, and B, are all in a line, the direction of AB, AP and PB are the same.
Third, seen in the diagram, Vectors OA and OB are known as Vectors "a" and "b" respectively.

Therefore, by knowing the statements above, one can find OP.

The first step of finding OP is what are it's closest or most relevant connections.
Point P lies on line AB, and both OA and OB are known, thus OP will likely be a combination of OA + AP or OB + BP.

Second, after finding a path towards OP, next is to find the Vector notation for AB, AP and/or PB.
AB can be found by AO + OB = -a + b = b - a. Therefore AP  =  (b - a)/4, and PB = (3/4)(b - a).
Thus OP can be found by OP = OA + AP  or OP = OB + BP = OB - PB.

The first method would be, OP = a + (b - a)/4 = b/4 + 3a/4.
The second method being, OP = b - (3/4)(b - a) = b/4 + 3a/4.


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