Three Dimensional Geometry - Algebra PYQ

\(A(-1,2-3), B(5,0,-6)\) 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

\(X\) 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

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 \(3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}\) ( resp. \(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}\) ). Then the shortest distance between the lines \(L_1\) and \(L_2\) is equal to

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

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

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

Let \(O\) be the origin and \(P\) be a point which is at a distance of 3 units from the origin. If the direction ratios of \(\overline{O P}\) are \((1,-2,-2)\), then the coordinates of \(P\) are

The direction cosines of the line joining the points \((-2,4,-5)\) and \((1,2,3)\) are

The equation of the plane passing through \(3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}\) and parallel to the vectors \(2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}\) and \(\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}\) is

The points (2, 3, 4), (\(-\)1, \(-\)2, 1) and (5, 8, 7) are

The sum of intercepts of the plane \(4 x+3 y+2 z=2\) on the coordinate axes is

If the lines, \(\frac{x-3}{2}=\frac{y-2}{3}=\frac{z-1}{\lambda}\) and \(\frac{x-2}{3}=\frac{y-3}{2}=\frac{z-2}{3}\) are coplanar, then \(\sin ^{-1}(\sin \lambda)+\cos ^{-1}(\cos \lambda)\) is equal to

If the vertices of the triangles are (1, 2, 3), (2, 3, 1), (3, 1, 2) and if H, G, S and I respectively denote its orthocentre, centroid, circumcentre and incentre, then H + G + S + I is equal to

A(2, 3, 4), B(4, 5, 7), C(2, \(-\)6, 3) and D(4, \(-\)4, k) are four points. If the line AB is parallel to CD, then k is equal to

The line passing through \((1,1,-1)\) and parallel to the vector \(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}\) meets the line \(\frac{x-3}{-1}=\frac{y+2}{5}=\frac{z-2}{-4}\) at \(A\) and the plane \(2 x-y+2 z+7=0\) at \(B\). Then \(A B\) is equal to

If the direction cosines of two lines are \(\left( {{2 \over 3},{2 \over 3},{1 \over 3}} \right)\) and \(\left( {{5 \over {13}},{{12} \over {13}},0} \right)\), then identify the direction ratios of a line which is bisecting one o the angle between them.

\(D, E, F\) are respectively the points on the sides \(B C, C A\) and \(A B\) of a \(\triangle A B C\) dividing them in the ratio \(2: 3,1: 2,3: 1\) internally. The lines \(\mathbf{B E}\) and \(\mathbf{C F}\) intersect on the line \(\mathbf{A D}\) at \(P\). If \(\mathbf{A P}=x_1 \cdot \mathbf{A} \mathbf{B}+y_1 \cdot \mathbf{A C}\), then \(x_1+y_1=\)

If the equation of the plane passing through the point \(A(-2,1,3)\) and perpendicular to the vector \(3 \hat{i}+\hat{j}+5 \hat{k}\) is \(a x+b y+c z+d=0\), then \(\frac{a+b}{c+d}=\)