If the planes bar{r} cdot(2 hat{i}-lambda hat | vedclass

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(A) Two planes $\bar{r} \cdot \bar{n}_1 = d_1$ and $\bar{r} \cdot \bar{n}_2 = d_2$ are parallel if their normal vectors $\bar{n}_1$ and $\bar{n}_2$ ar

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$\frac{1}{2}, 2$

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(A) Two planes $\bar{r} \cdot \bar{n}_1 = d_1$ and $\bar{r} \cdot \bar{n}_2 = d_2$ are parallel if their normal vectors $\bar{n}_1$ and $\bar{n}_2$ are proportional,i.e.,$\bar{n}_1 = k \bar{n}_2$. Given normal vectors are $\bar{n}_1 = 2 \hat{i} - \lambda \hat{j} + \hat{k}$ and $\bar{n}_2 = 4 \hat{i} - \hat{j} + \mu \hat{k}$. For the planes to be parallel,the ratios of the components of the normal vectors must be equal: $\frac{2}{4} = \frac{-\lambda}{-1} = \frac{1}{\mu}$ From $\frac{2}{4} = \frac{-\lambda}{-1}$,we get $\frac{1}{2} = \lambda$,so $\lambda = \frac{1}{2}$. From $\frac{2}{4} = \frac{1}{\mu}$,we get $\frac{1}{2} = \frac{1}{\mu}$,so $\mu = 2$. Thus,the values are $\lambda = \frac{1}{2}$ and $\mu = 2$.

If the planes $\bar{r} \cdot(2 \hat{i}-\lambda \hat{j}+\hat{k})=3$ and $\bar{r} \cdot(4 \hat{i}-\hat{j}+\mu \hat{k})=5$ are parallel,then the values of $\lambda$ and $\mu$ are respectively:

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