A) \[\left( \frac{2}{3},\frac{4}{3},\frac{2}{3} \right)\]
B) \[\left( -\frac{2}{3},-\frac{4}{3},\frac{2}{3} \right)\]
C) \[\left( \frac{2}{3},\frac{4}{3},-\frac{2}{3} \right)\]
D) None of these
Correct Answer: A
Solution :
[a] Let the point P be (x, y, z), then the vector \[x\hat{i}+y\hat{j}+z\hat{k}\] Will lie on the line. Thus. \[(x-1)\hat{i}+(y-1)\hat{j}+(z-1)\hat{k}=-\lambda \hat{i}+\lambda \hat{j}-\lambda \hat{k}\] \[\Rightarrow x=1-\lambda ,y=1+\lambda \] and \[z=1-\lambda \] Now point P is nearest to the origin \[\Rightarrow D={{(1-\lambda )}^{2}}+{{(1+\lambda )}^{2}}+{{(1-\lambda )}^{2}}\] Or \[\frac{dD}{d\lambda }=-4(1-\lambda )+2(1+\lambda )=0\]\[\Rightarrow \lambda =\frac{1}{3}\] Hence, the point is \[\left( \frac{2}{3},\frac{4}{3},\frac{2}{3} \right)\]
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