The equation of the line through the point (0, 1, 2) and perpendicular to the line {{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over { - 2}} is :

Let $$\alpha$$ be the angle between the lines whose direction cosines satisfy the equations l + m $$-$$ n = 0 and l2 + m2 $$-$$ n2 = 0. Then the value of sin4$$\alpha$$ + cos4$$\alpha$$ is :

A plane passes through the points A(1, 2, 3), B(2, 3, 1) and C(2, 4, 2). If O is the origin and P is (2, $$-$$1, 1), then the projection of $$\overrightarrow{OP}$$ on this plane is of length :

Consider the three planes P1 : 3x + 15y + 21z = 9, P2 : x $$-$$ 3y $$-$$ z = 5, and P3 : 2x + 10y + 14z = 5 Then, which one of the following is true?

If (1, 5, 35), (7, 5, 5), (1, $$\lambda$$, 7) and (2$$\lambda$$, 1, 2) are coplanar, then the sum of all possible values of $$\lambda$$ is :

If the mirror image of the point (1, 3, 5) with respect to the plane 4x $$-$$ 5y + 2z = 8 is ($$\alpha$$, $$\beta$$, $$\gamma$$), then 5($$\alpha$$ + $$\beta$$ + $$\gamma$$) equals :

Let L be a line obtained from the intersection of two planes x + 2y + z = 6 and y + 2z = 4. If point P($$\alpha$$, $$\beta$$, $$\gamma$$) is the foot of perpendicular from (3, 2, 1) on L, then the value of 21($$\alpha$$ + $$\beta$$ + $$\gamma$$) equals :

Let the position vectors of two points P and Q be 3$$\hat{i}$$ $$-$$ $$\hat{j}$$ + 2$$\hat{k}$$ and $$\hat{i}$$ + 2$$\hat{j}$$ $$-$$ 4$$\hat{k}$$, respectively. Let R and S be two points such that the direction ratios of lines PR and QS are (4, $$-$$1, 2) and ($$-$$2, 1, $$-$$2), respectively. Let lines PR and QS intersect at T. If the vector $$\overrightarrow{TA}$$ is perpendicular to both $$\overrightarrow{PR}$$ and $$\overrightarrow{QS}$$ and the length of vector $$\overrightarrow{TA}$$ is $$\sqrt{5}$$ units, then the modulus of a position vector of A is :

Let P be a plane lx + my + nz = 0 containing the line, {{1 - x} \over 1} = {{y + 4} \over 2} = {{z + 2} \over 3}. If plane P divides the line segment AB joining points A($$-$$3, $$-$$6, 1) and B(2, 4, $$-$$3) in ratio k : 1 then the value of k is equal to :

If for a > 0, the feet of perpendiculars from the points A(a, $$-$$2a, 3) and B(0, 4, 5) on the plane lx + my + nz = 0 are points C(0, $$-$$a, $$-$$1) and D respectively, then the length of line segment CD is equal to :

If the foot of the perpendicular from point (4, 3, 8) on the line {L_1}:{{x - a} \over l} = {{y - 2} \over 3} = {{z - b} \over 4}, l $$\ne$$ 0 is (3, 5, 7), then the shortest distance between the line L1 and line {L_2}:{{x - 2} \over 3} = {{y - 4} \over 4} = {{z - 5} \over 5} is equal to :

If (x, y, z) be an arbitrary point lying on a plane P which passes through the points (42, 0, 0), (0, 42, 0) and (0, 0, 42), then the value of the expression 3 + {{x - 11} \over {{{(y - 19)}^2}{{(z - 12)}^2}}} + {{y - 19} \over {{{(x - 11)}^2}{{(z - 12)}^2}}} + {{z - 12} \over {{{(x - 11)}^2}{{(y - 19)}^2}}} - {{x + y + z} \over {14(x - 11)(y - 19)(z - 12)}} is equal to :

The equation of the plane which contains the y-axis and passes through the point (1, 2, 3) is :

If the equation of plane passing through the mirror image of a point (2, 3, 1) with respect to line {{x + 1} \over 2} = {{y - 3} \over 1} = {{z + 2} \over { - 1}} and containing the line {{x - 2} \over 3} = {{1 - y} \over 2} = {{z + 1} \over 1} is $$\alpha$$x + $$\beta$$y + $$\gamma$$z = 24, then $$\alpha$$ + $$\beta$$ + $$\gamma$$ is equal to :

The lines x = ay $$-$$ 1 = z $$-$$ 2 and x = 3y $$-$$ 2 = bz $$-$$ 2, (ab $$\ne$$ 0) are coplanar, if :

Consider the line L given by the equation {{x - 3} \over 2} = {{y - 1} \over 1} = {{z - 2} \over 1}. Let Q be the mirror image of the point (2, 3, $$-$$1) with respect to L. Let a plane P be such that it passes through Q, and the line L is perpendicular to P. Then which of the following points is on the plane P?

Let L be the line of intersection of planes $$\overrightarrow{r}.(\hat{i}-\hat{j}+2\hat{k})=2$$ and $$\overrightarrow{r}.(2\hat{i}+\hat{j}-\hat{k})=2$$. If $$P(\alpha ,\beta ,\gamma )$$ is the foot of perpendicular on L from the point (1, 2, 0), then the value of $$35(\alpha +\beta +\gamma )$$ is equal to :

If the shortest distance between the straight lines $$3(x-1)=6(y-2)=2(z-1)$$ and $$4(x-2)=2(y-\lambda )=(z-3),\lambda \in R$$ is {1 \over {\sqrt {38} }}, then the integral value of $$\lambda$$ is equal to :

Let the foot of perpendicular from a point P(1, 2, $$-$$1) to the straight line L:{x \over 1} = {y \over 0} = {z \over { - 1}} be N. Let a line be drawn from P parallel to the plane x + y + 2z = 0 which meets L at point Q. If $$\alpha$$ is the acute angle between the lines PN and PQ, then cos$$\alpha$$ is equal to ________________.

For real numbers $$\alpha$$ and $$\beta$$ $$\ne$$ 0, if the point of intersection of the straight lines {{x - \alpha } \over 1} = {{y - 1} \over 2} = {{z - 1} \over 3} and {{x - 4} \over \beta } = {{y - 6} \over 3} = {{z - 7} \over 3}, lies on the plane x + 2y $$-$$ z = 8, then $$\alpha$$ $$-$$ $$\beta$$ is equal to :