If the position vectors of points P, Q, R, an | vedclass

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(A) The position vector of $\overline{PQ} = 	ext{P.V. of } Q - 	ext{P.V. of } P = (\hat{i} + 2\hat{j} + 3\hat{k}) - (2\hat{i} + 3\hat{j} + 5\hat{k})

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$\overline{PQ} \parallel \overline{RS}$

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(A) The position vector of $\overline{PQ} = 	ext{P.V. of } Q - 	ext{P.V. of } P = (\hat{i} + 2\hat{j} + 3\hat{k}) - (2\hat{i} + 3\hat{j} + 5\hat{k}) = -\hat{i} - \hat{j} - 2\hat{k}$.The position vector of $\overline{RS} = 	ext{P.V. of } S - 	ext{P.V. of } R = (\hat{i} + 10\hat{j} + 10\hat{k}) - (-5\hat{i} + 4\hat{j} - 2\hat{k}) = 6\hat{i} + 6\hat{j} + 12\hat{k}$.For $\overline{PQ} \parallel \overline{RS}$,the ratios of their components must be equal:$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.Substituting the values: $\frac{-1}{6} = \frac{-1}{6} = \frac{-2}{12}$.Simplifying the fractions: $-\frac{1}{6} = -\frac{1}{6} = -\frac{1}{6}$.Since the ratios are equal,the vectors are parallel. Therefore,$\overline{PQ} \parallel \overline{RS}$.

If the position vectors of points $P, Q, R,$ and $S$ are $2\hat{i} + 3\hat{j} + 5\hat{k}$,$\hat{i} + 2\hat{j} + 3\hat{k}$,$-5\hat{i} + 4\hat{j} - 2\hat{k}$,and $\hat{i} + 10\hat{j} + 10\hat{k}$ respectively,then:

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