The point of intersection of the lines repres | vedclass

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Question

(A) Let the point of intersection be $P$. The coordinates of any point on the first line are $(1+2\lambda, 2+3\lambda, -1+4\lambda)$ and on the second

Vedclass · Accepted Answer

$3 \hat{i}+5 \hat{j}+3 \hat{k}$

Vedclass · Answer

(A) Let the point of intersection be $P$. The coordinates of any point on the first line are $(1+2\lambda, 2+3\lambda, -1+4\lambda)$ and on the second line are $(-1+\mu, -3+2\mu, 7-\mu)$.Equating the coordinates,we get:$1+2\lambda = -1+\mu \implies 2\lambda - \mu = -2$ ... $(i)$$2+3\lambda = -3+2\mu \implies 3\lambda - 2\mu = -5$ ... $(ii)$$-1+4\lambda = 7-\mu \implies 4\lambda + \mu = 8$ ... $(iii)$Adding equations $(i)$ and $(iii)$:$(2\lambda - \mu) + (4\lambda + \mu) = -2 + 8$$6\lambda = 6 \implies \lambda = 1$Substituting $\lambda = 1$ in equation $(i)$:$2(1) - \mu = -2 \implies \mu = 4$Checking these values in equation $(ii)$:$3(1) - 2(4) = 3 - 8 = -5$,which is correct.Substituting $\lambda = 1$ in the first line equation:$r = (\hat{i}+2 \hat{j}-\hat{k}) + 1(2 \hat{i}+3 \hat{j}+4 \hat{k}) = 3 \hat{i}+5 \hat{j}+3 \hat{k}$.Thus,the point of intersection is $3 \hat{i}+5 \hat{j}+3 \hat{k}$.

The point of intersection of the lines represented by $r=(\hat{i}+2 \hat{j}-\hat{k})+\lambda(2 \hat{i}+3 \hat{j}+4 \hat{k})$ and $r=(-\hat{i}-3 \hat{j}+7 \hat{k})+\mu(\hat{i}+2 \hat{j}-\hat{k})$ is

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