The image point of $(5, 4, 6)$ in the plane $x + y + 2z - 15 = 0$ is

Consider the tetrahedron with the vertices $A(3,2,4)$,$B(x_1, y_1, 0)$,$C(x_2, y_2, 0)$,and $D(x_3, y_3, 0)$. If the triangle $BCD$ is formed by the lines $y=x$,$x+y=6$,and $y=1$,then the centroid of the tetrahedron is

Find the image of the point $(1, 2, 3)$ in the line $\vec{r} = (6\hat{i} + 7\hat{j} + 7\hat{k}) + \lambda(3\hat{i} + 2\hat{j} - 2\hat{k})$.

$AB$ and $BC$ are diagonals of adjacent faces of a rectangular box with its center at the origin,and its edges parallel to the coordinate axes. If the angles $\angle BOC, \angle COA$,and $\angle AOB$ are $\alpha, \beta$,and $\gamma$ respectively,then $\cos \alpha + \cos \beta + \cos \gamma$ is equal to:

If $A(0,1,2)$,$B(2,-1,3)$,and $C(1,-3,1)$ are the vertices of a triangle,then the distance between its circumcentre and orthocentre is

If a line makes angles $\alpha, \beta, \gamma, \delta$ with the four diagonals of a cube,then find the value of $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma + \cos^2 \delta$.