The constant value (lambda + mu) for which th | vedclass

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(D) For the lines to intersect,there must exist values of $\lambda$ and $\mu$ such that the coordinates of any point on the lines are equal.The first

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(D) For the lines to intersect,there must exist values of $\lambda$ and $\mu$ such that the coordinates of any point on the lines are equal.The first line is $\vec{r} = (2 + \lambda)\hat{i} + (1 - 2\lambda)\hat{j} + 1\hat{k}$.The second line is $\vec{r} = 1\hat{i} + (1 + \mu)\hat{j} + (-3 + 2\mu)\hat{k}$.Equating the components:$1) 2 + \lambda = 1 \Rightarrow \lambda = -1$$2) 1 - 2\lambda = 1 + \mu$$3) 1 = -3 + 2\mu$From equation $(3)$,$2\mu = 4 \Rightarrow \mu = 2$.Check consistency with equation $(2)$:$1 - 2(-1) = 1 + 2 \Rightarrow 1 + 2 = 3 \Rightarrow 3 = 3$. This is consistent.Thus,$\lambda = -1$ and $\mu = 2$.The value of $(\lambda + \mu) = -1 + 2 = 1$.

The constant value $(\lambda + \mu)$ for which the lines $\vec{r} = (2\hat{i} + \hat{j} + \hat{k}) + \lambda(\hat{i} - 2\hat{j})$ and $\vec{r} = (\hat{i} + \hat{j} - 3\hat{k}) + \mu(\hat{j} + 2\hat{k})$ intersect each other is equal to (where $\lambda$ and $\mu$ are parameters).

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