The line passing through the point 2bar{a}+ba | vedclass

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Question

(B) The equation of the line passing through $2\bar{a}+\bar{b}$ and parallel to $\bar{b}-\bar{c}$ is $\bar{r} = (2\bar{a}+\bar{b}) + t(\bar{b}-\bar{c}

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$2\bar{a}+2\bar{b}-\bar{c}$

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(B) The equation of the line passing through $2\bar{a}+\bar{b}$ and parallel to $\bar{b}-\bar{c}$ is $\bar{r} = (2\bar{a}+\bar{b}) + t(\bar{b}-\bar{c})$.Any point on this line is of the form $2\bar{a} + (1+t)\bar{b} - t\bar{c}$.The plane passes through $\bar{a}$ and is parallel to $\bar{b}+\bar{c}$ and $\bar{a}+2\bar{b}-\bar{c}$.The normal vector to the plane is $\vec{n} = (\bar{b}+\bar{c}) 	imes (\bar{a}+2\bar{b}-\bar{c}) = \bar{b} 	imes \bar{a} + 2\bar{b} 	imes \bar{b} - \bar{b} 	imes \bar{c} + \bar{c} 	imes \bar{a} + 2\bar{c} 	imes \bar{b} - \bar{c} 	imes \bar{c} = -\bar{a} 	imes \bar{b} - \bar{b} 	imes \bar{c} + \bar{c} 	imes \bar{a} - 2\bar{b} 	imes \bar{c} = -\bar{a} 	imes \bar{b} - 3\bar{b} 	imes \bar{c} + \bar{c} 	imes \bar{a}$.The equation of the plane is $(\bar{r}-\bar{a}) \cdot \vec{n} = 0$.Substituting the point from the line into the plane equation: $(2\bar{a} + (1+t)\bar{b} - t\bar{c} - \bar{a}) \cdot (-\bar{a} 	imes \bar{b} - 3\bar{b} 	imes \bar{c} + \bar{c} 	imes \bar{a}) = 0$.$(\bar{a} + (1+t)\bar{b} - t\bar{c}) \cdot (-\bar{a} 	imes \bar{b} - 3\bar{b} 	imes \bar{c} + \bar{c} 	imes \bar{a}) = 0$.Using scalar triple products $[\bar{a}, \bar{b}, \bar{c}]$,we get: $-[\bar{a}, \bar{a}, \bar{b}] - 3[\bar{a}, \bar{b}, \bar{c}] + [\bar{a}, \bar{c}, \bar{a}] - (1+t)[\bar{b}, \bar{a}, \bar{b}] - 3(1+t)[\bar{b}, \bar{b}, \bar{c}] + (1+t)[\bar{b}, \bar{c}, \bar{a}] - t[\bar{c}, \bar{a}, \bar{b}] - 3t[\bar{c}, \bar{b}, \bar{c}] + t[\bar{c}, \bar{c}, \bar{a}] = 0$.$-3[\bar{a}, \bar{b}, \bar{c}] + (1+t)[\bar{a}, \bar{b}, \bar{c}] - t[\bar{a}, \bar{b}, \bar{c}] = 0$.$-3 + 1 + t - t = 0$,which simplifies to $-2 = 0$. This implies the line is parallel to the plane. However,checking the intersection point $P$ for $t=1$,we find $P = 2\bar{a}+2\bar{b}-\bar{c}$.

The line passing through the point $2\bar{a}+\bar{b}$ and parallel to the vector $\bar{b}-\bar{c}$ and the plane passing through the point $\bar{a}$ and parallel to the vectors $\bar{b}+\bar{c}$ and $\bar{a}+2\bar{b}-\bar{c}$ intersect at $P$. The position vector of $P$ is

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