The position vectors of two points A and B ar | vedclass

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(A) Let the position vectors of $A$ and $B$ be $\vec{a} = \bar{i}+2\bar{j}+3\bar{k}$ and $\vec{b} = 7\bar{i}-\bar{k}$.Let $P$ divide $AB$ in the ratio

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$6$

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(A) Let the position vectors of $A$ and $B$ be $\vec{a} = \bar{i}+2\bar{j}+3\bar{k}$ and $\vec{b} = 7\bar{i}-\bar{k}$.Let $P$ divide $AB$ in the ratio $m:n$. Then $\vec{p} = \frac{m\vec{b} + n\vec{a}}{m+n}$.$-2\bar{i}+3\bar{j}+5\bar{k} = \frac{m(7\bar{i}-\bar{k}) + n(\bar{i}+2\bar{j}+3\bar{k})}{m+n}$.Comparing coefficients: $x: -2(m+n) = 7m + n \implies 9m = -3n \implies m/n = -1/3$.Thus,$P$ divides $AB$ externally in the ratio $1:3$.The harmonic conjugate $Q$ divides $AB$ internally in the same ratio $1:3$.$\vec{q} = \frac{1\vec{b} + 3\vec{a}}{1+3} = \frac{(7\bar{i}-\bar{k}) + 3(\bar{i}+2\bar{j}+3\bar{k})}{4} = \frac{10\bar{i}+6\bar{j}+8\bar{k}}{4} = 2.5\bar{i}+1.5\bar{j}+2\bar{k}$.The sum of the scalar components is $2.5 + 1.5 + 2 = 6$.

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