# Given the vectors a and b: a = 4i + 7j b = 10i + 10j Calculate, using a vector method, the vector c if AB : BC is in the ratio 3 : 2

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The position vectors of A is 4i+7j.

The position vector of the point C is 10i+10j.

We want the position vector of a point P which divides the vector AB in the ratio 3:2.

We know that the position vector of a point Qthat divides the position vector P(x1i+y1j) and R(x2i+y2j) in m: n ratio is (nx1+mx2)i/(m+n)+(ny1+my2)j/(m+n).

So the position vector of B that divides the position vectors A(4i+7j) and C(10i+10j) is B =(2*4+3*10)i/(3+2)+(2*7+3*10)j/(3+2)}.

So B = (38/5)i+(54/5)j = 7.6i+10.8j.

Therefore the position vector of B is 7.6i+10.8j which divides the position vector of A and C in the ratio 3:2.