Angle between the planes, $\mathbf{r} \cdot(12 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})=5$ and, $\mathbf{r} \cdot(5 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})=7$ is

The shortest distance between the skew lines $\mathbf{r}=(2 \hat{\mathbf{i}}-\hat{\mathbf{j}})+t(\hat{\mathbf{i}}+2 \hat{\mathbf{k}})$ and $\mathbf{r}=(-2 \hat{\mathbf{i}}+\hat{\mathbf{k}})+s(\hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}})$ is

If the plane $x-y+z+4=0$ divides the line joining the points $P(2,3,-1)$ and $Q(1,4,-2)$ in the ratio $l: m$, then $l+m$ is

If the line with direction ratios $(1, \alpha, \beta)$ is perpendicular to the line with direction ratios $(-1,2,1)$ and parallel to the line with direction ratios $(\alpha, 1, \beta)$ then $(\alpha, \beta)$ is