Three Dimensional Geometry

The direction cosines of a line which makes
equal angles with the coordinate axes are

The ratio in which the YZ-plane divides the line joining (2, 4, 5) and (3, 5, \(-\)4) is

Find the equation of the plane passing through the point \((2,1,3)\) and perpendicular to the planes \(x-2 y+2 z+3=0\) and \(3 x-2 y+4 z-4=0\).

If the projections of the line segment AB on xy, yz and zx planes are \(\sqrt{15},\sqrt{46},7\) respectively, then the projection of AB on Y-axis is

\(A(-1,2-3), B(5,0,-6)\) and \(C(0,4,-1)\) are the vertices of a \(\triangle A B C\). The direction cosines of internal bisector of \(\angle B A C\) are

If the points (2, 4, \(-\)1), (3, 6, \(-\)1) and (4, 5, \(-\)1) are three consecutive vertices of a parallelogram,
then its fourth vertex is

Let \(L_1\) (resp, \(L_2\) ) be the line passing through \(2 \hat{\mathbf{i}}-\hat{\mathbf{k}}\) (resp. \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-3 \hat{\mathbf{k}})\) and parallel to \(3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) (...

\(X\) intercept of the plane containing the line of intersection of the planes \(x-2 y+z+2=0\) and \(3 x-y-z+1=0\) and also passing through \((1,1,1)\) is