Find the position vector of the point in space P that lies on the line AB such that with m,neR. Find the position vector of P if A(1,2,1), B(3,-1,2), m=3 and n =2.

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We are given two points A(1,2,1) and B(3,−1,2) in space, and we are asked to find the position vector of the point P on the line passing through A and B, such that |AP|:|PB|=m:n=3:2.Let 𝐚 and 𝐛 be the position vectors of points A and B, respectively, and let 𝐝 be the direction vector of the line passing through A and B. Then we have:𝐚=(121),𝐛=(3−12),𝐝=𝐛−𝐚=(2−31)Let 𝐩 be the position vector of point P on the line. Then we can express 𝐩 as:𝐩=𝐚+t𝐝for some scalar t. We want to find t such that |AP|:|PB|=3:2. Let |AP|=3k and |PB|=2k for some scalar k. Then we have:|AP|=|𝐩−𝐚|=|𝐚+t𝐝−𝐚|=|t𝐝|=t|𝐝|=3k|PB|=|𝐛−𝐩|=|𝐛−𝐚−t𝐝|=|𝐝−t𝐝|=(1−t)|𝐝|=2kSolving these equations for t and k, we get:t=513,k=313Substituting this value of t into the expression for 𝐩, we get:𝐩=(121)+513(2−31)=(2713−113813)Therefore, the position vector of point P is 𝐩=(2713−113813).

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