If the straight lines $\vec{r} = (1, 2, 3) + k(\lambda, 2, 3), k \in R$ and $\vec{r} = (2, 3, 1) + k(3, \lambda, 2), k \in R$ intersect at a point,then the integer $\lambda$ is equal to:

If the square of the shortest distance between the lines $\frac{x-2}{1}=\frac{y-1}{2}=\frac{z+3}{-3}$ and $\frac{x+1}{2}=\frac{y+3}{4}=\frac{z+5}{-5}$ is $\frac{m}{n}$,where $m, n$ are coprime numbers,then $m+n$ is equal to:

The position vector of the point of intersection of the line joining the points $2 \hat{i}-\hat{j}+6 \hat{k}$ and $3 \hat{i}-\hat{j}-7 \hat{k}$,and the line joining the points $2 \hat{i}+\hat{j}-6 \hat{k}$ and $3 \hat{i}-\hat{j}-7 \hat{k}$ is:

Two lines $L_1: x=5, \frac{y}{3-\alpha}=\frac{z}{-2}$ and $L_2: x=\alpha, \frac{y}{-1}=\frac{z}{2-\alpha}$ are coplanar. Then $\alpha$ can take value$(s)$.

$P, Q, R$ and $S$ are four points with the position vectors $3i-4j+5k, 0i+0j+4k, -4i+5j+1k$ and $-3i+4j+3k$,respectively. Then,the line $PQ$ meets the line $RS$ at the point

The line joining the points $(-2, 1, -8)$ and $(a, b, c)$ is parallel to the line whose direction ratios are $6, 2, 3$. The values of $a, b, c$ are