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14 Oct 2011 16:07:49 IST

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equation of plane passing through points a b and c not in same line is r=ma+nb+pc/m+n+p

Mathematics

show tthat the equation of plane passing through thre given points A,Band Cnot inthe same straight line and having position vectors a b and c relative to an origin o can be written as r= ma+nb+pc/m+n+p,where m n and p are scalers

also verify that the equation is independent of origin.

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Sumit Bhattacharjee

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Joined: 18 Oct 2011

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18 Oct 2011 22:37:35 IST

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Suppose the points A, B, C all lie on the plane, andOA=a, OB=b, OC=c.Now the vector AB=b-a lies in the plane. Similarly, AC=c-a, lies in the plane. now if P is any point in the plane with position vector, r, then OP=OA+AP or r=a+AP. By construction we can make, AP=AD+DP, where D is on AC, produced such that DP is parallel to AB. So if AB=nAC, DP=mAB, for some parameters m and n, then AP=mAB+nAC=m(b-a)+n(c-a). Finally we can write the equation as, r=a+m(b-a)+n(c-a), or r=(1-m-n)a+mb+nc, or, r=(1-m-n)a+mb+nc/{(1-m-n)+m+n}, or in the form, r=ma+nb+pc/m+n+p

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