If the line joining the points $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ and $\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ intersects the plane passing through the points $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}, 2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and $\hat{\mathbf{k}}-2 \hat{\mathbf{i}}$ at $\mathbf{r}$, then $\mathbf{r} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}})=$

The vector equation of a plane passing through the line of intersection of the planes $\mathbf{r} \cdot(\hat{\mathbf{i}}-2 \hat{\mathbf{k}})=3, \mathbf{r} \cdot(2 \hat{\mathbf{j}}+\hat{\mathbf{k}})=5$ and the point $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ is

The points $A(-1,2,3), B(2,-3,1)$ and $C(3,1,-2)$

The directions cosines of the line making angles $\frac{\pi}{4}, \frac{\pi}{3}$ and $\theta\left(0<\theta<\frac{\pi}{2}\right)$ respectively with $X, Y$ and $Z$ axes are