Let vec{a}, vec{b}, vec{c} be three non-copla | vedclass

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(B) The line $L$ passes through $P_1 = \vec{a}-\vec{b}+\vec{c}$ and $P_2 = \vec{b}-\vec{c}$. The direction vector of $L$ is $\vec{v} = P_2 - P_1 = (\v

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$\vec{b}+\vec{c}$

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(B) The line $L$ passes through $P_1 = \vec{a}-\vec{b}+\vec{c}$ and $P_2 = \vec{b}-\vec{c}$. The direction vector of $L$ is $\vec{v} = P_2 - P_1 = (\vec{b}-\vec{c}) - (\vec{a}-\vec{b}+\vec{c}) = -\vec{a} + 2\vec{b} - 2\vec{c}$.Thus, the equation of line $L$ is $\vec{r} = (\vec{a}-\vec{b}+\vec{c}) + \lambda(-\vec{a} + 2\vec{b} - 2\vec{c})$.The plane $\pi$ passes through $A = 2\vec{a}-\vec{b}$, $B = 2\vec{b}-\vec{c}$, and $C = 2\vec{c}-\vec{a}$.The normal vector to the plane is $\vec{n} = (B-A) 	imes (C-A) = (-2\vec{a}+3\vec{b}-\vec{c}) 	imes (-3\vec{a}+\vec{b}+2\vec{c}) = 7(\vec{a} 	imes \vec{b} + \vec{b} 	imes \vec{c} + \vec{c} 	imes \vec{a})$.The equation of the plane is $(\vec{r} - (2\vec{a}-\vec{b})) \cdot (\vec{a} 	imes \vec{b} + \vec{b} 	imes \vec{c} + \vec{c} 	imes \vec{a}) = 0$.Substituting $\vec{r} = \vec{a}-\vec{b}+\vec{c} + \lambda(-\vec{a} + 2\vec{b} - 2\vec{c})$ into the plane equation and solving for $\lambda$, we find $\lambda = 1$.Substituting $\lambda = 1$ into the line equation gives $\vec{r} = (\vec{a}-\vec{b}+\vec{c}) + 1(-\vec{a} + 2\vec{b} - 2\vec{c}) = \vec{b}-\vec{c}$.

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