The line joining the points 6 overrightarrow{ | vedclass

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(D) Let the two lines be $L_1$ and $L_2$. The line $L_1$ passes through $A = 6 \overrightarrow{a}-4 \overrightarrow{b}+4 \overrightarrow{c}$ and $B =

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(D) Let the two lines be $L_1$ and $L_2$. The line $L_1$ passes through $A = 6 \overrightarrow{a}-4 \overrightarrow{b}+4 \overrightarrow{c}$ and $B = -4 \overrightarrow{c}$. The direction vector of $L_1$ is $\overrightarrow{d_1} = B - A = -6 \overrightarrow{a}+4 \overrightarrow{b}-8 \overrightarrow{c}$. The equation of $L_1$ is $\overrightarrow{r} = 6 \overrightarrow{a}-4 \overrightarrow{b}+4 \overrightarrow{c} + m(-6 \overrightarrow{a}+4 \overrightarrow{b}-8 \overrightarrow{c})$ ... $(i)$.The line $L_2$ passes through $C = -\overrightarrow{a}-2 \overrightarrow{b}-3 \overrightarrow{c}$ and $D = \overrightarrow{a}+2 \overrightarrow{b}-5 \overrightarrow{c}$. The direction vector of $L_2$ is $\overrightarrow{d_2} = D - C = 2 \overrightarrow{a}+4 \overrightarrow{b}-2 \overrightarrow{c}$. The equation of $L_2$ is $\overrightarrow{r} = -\overrightarrow{a}-2 \overrightarrow{b}-3 \overrightarrow{c} + n(2 \overrightarrow{a}+4 \overrightarrow{b}-2 \overrightarrow{c})$ ... $(ii)$.Equating the two expressions for $\overrightarrow{r}$:$6 \overrightarrow{a}-4 \overrightarrow{b}+4 \overrightarrow{c} + m(-6 \overrightarrow{a}+4 \overrightarrow{b}-8 \overrightarrow{c}) = -\overrightarrow{a}-2 \overrightarrow{b}-3 \overrightarrow{c} + n(2 \overrightarrow{a}+4 \overrightarrow{b}-2 \overrightarrow{c})$Comparing coefficients of $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$:$6 - 6m = -1 + 2n \implies 6m + 2n = 7$ ... $(iii)$$-4 + 4m = -2 + 4n \implies 4m - 4n = 2 \implies 2m - 2n = 1$ ... $(iv)$$4 - 8m = -3 - 2n \implies 8m - 2n = 7$ ... $(v)$Adding $(iii)$ and $(iv)$: $8m = 8 \implies m = 1$. Substituting $m=1$ in $(iv)$: $2(1) - 2n = 1 \implies 2n = 1 \implies n = 1/2$. Checking in $(v)$: $8(1) - 2(1/2) = 8 - 1 = 7$. The values satisfy the system.Substituting $m=1$ in $(i)$: $\overrightarrow{r} = 6 \overrightarrow{a}-4 \overrightarrow{b}+4 \overrightarrow{c} - 6 \overrightarrow{a}+4 \overrightarrow{b}-8 \overrightarrow{c} = -4 \overrightarrow{c}$.

The line joining the points $6 \overrightarrow{a}-4 \overrightarrow{b}+4 \overrightarrow{c}$ and $-4 \overrightarrow{c}$ and the line joining the points $-\overrightarrow{a}-2 \overrightarrow{b}-3 \overrightarrow{c}$ and $\overrightarrow{a}+2 \overrightarrow{b}-5 \overrightarrow{c}$ intersect at:

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