The point of intersection of the lines represented by $\mathbf{r}=(\hat{\mathbf{i}}+2…
The point of intersection of the lines represented by $\mathbf{r}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})$ and $\mathbf{r}=(-\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})$ is
Let point $P$ is the point of intersection of given lines $\mathbf{r}=(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})$ and $\mathbf{r}=(-\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+7 \hat{\mathbf{k}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}})$
So, for point $P$,
From Eqs. (i) and (iii), we get $\lambda=1$ and $\mu=4$, which satisfy the Eq. (ii), so required point of intersection $P$ is $3 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$. Hence, option (a) is correct.
