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(C) The equation of a plane passing through the intersection of two planes $P_1: \overrightarrow{r} \cdot \overrightarrow{n_1} = d_1$ and $P_2: \overr

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$\overrightarrow{r} \cdot (\hat{i} + 7\hat{j} + 3\hat{k}) = 7$

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(C) The equation of a plane passing through the intersection of two planes $P_1: \overrightarrow{r} \cdot \overrightarrow{n_1} = d_1$ and $P_2: \overrightarrow{r} \cdot \overrightarrow{n_2} = d_2$ is given by $(\overrightarrow{r} \cdot \overrightarrow{n_1} - d_1) + \lambda(\overrightarrow{r} \cdot \overrightarrow{n_2} - d_2) = 0$.Given planes are $\overrightarrow{r} \cdot (\hat{i} + \hat{j} + \hat{k}) - 1 = 0$ and $\overrightarrow{r} \cdot (\hat{i} - 2\hat{j}) + 2 = 0$.The equation of the required plane is $\overrightarrow{r} \cdot (\hat{i} + \hat{j} + \hat{k}) - 1 + \lambda(\overrightarrow{r} \cdot (\hat{i} - 2\hat{j}) + 2) = 0$.Rearranging the terms,we get $\overrightarrow{r} \cdot [\hat{i}(1 + \lambda) + \hat{j}(1 - 2\lambda) + \hat{k}(1)] = 1 - 2\lambda$.Since the plane passes through the point $(1, 0, 2)$,the position vector is $\overrightarrow{r} = \hat{i} + 2\hat{k}$.Substituting this into the equation: $(\hat{i} + 2\hat{k}) \cdot [\hat{i}(1 + \lambda) + \hat{j}(1 - 2\lambda) + \hat{k}(1)] = 1 - 2\lambda$.$(1 + \lambda) + 2(1) = 1 - 2\lambda$.$3 + \lambda = 1 - 2\lambda$.$3\lambda = -2 \implies \lambda = -\frac{2}{3}$.Substituting $\lambda = -\frac{2}{3}$ back into the equation: $\overrightarrow{r} \cdot [\hat{i}(1 - \frac{2}{3}) + \hat{j}(1 + \frac{4}{3}) + \hat{k}(1)] = 1 - 2(-\frac{2}{3})$.$\overrightarrow{r} \cdot [\hat{i}(\frac{1}{3}) + \hat{j}(\frac{7}{3}) + \hat{k}] = 1 + \frac{4}{3} = \frac{7}{3}$.Multiplying by $3$,we get $\overrightarrow{r} \cdot (\hat{i} + 7\hat{j} + 3\hat{k}) = 7$.

The vector equation of the plane passing through the intersection of the planes $\overrightarrow{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$ and $\overrightarrow{r} \cdot (\hat{i} - 2\hat{j}) = -2$,and the point $(1, 0, 2)$ is:

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