Three Dimension...

If $$\alpha ,\beta ,\gamma $$ are the angles which a directed line makes with the positive directions of the coordinate axes, then $$\sin ^{ 2 }{ \alpha  } +\sin ^{ 2 }{ \beta  } +\sin ^{ 2 }{ \gamma  } $$ is equal to

The acute angle between the lines whose direction ratios are given by $$l+m-n=0$$ and $${ l }^{ 2 }+{ m }^{ 2 }-{ n }^{ 2 }=0$$, is

ABCD is a trapezium in which AB and CD are parallel sides. If $$l(AB) =  3 l (CD) $$ and $$\bar {DC} = 2 \hat {i} - 5 \hat {k}$$. Then vector  $$ \bar {AB} $$ is

If a line makes $${ 45 }^{ o },{ 60 }^{ o }$$ with positive direction of axes $$x$$ and $$y$$ then the angle it makes with the z-axis is:

The dc's $$(l,m,n)$$ of two lines are connected between the relation $$l + m + n = 0, \ lm = 0$$, then the angle between the lines is 

$$A = \begin{bmatrix} l_{1}& m_{1} & n_{1}\\ l_{2} & m_{2} & n_{2}\\ l_{3} & m_{3} & n_{3}\end{bmatrix}$$ and $$B = \begin{bmatrix}p_{1} & q_{1} & r_{1}\\ p_{2} & q_{2} & r_{2}\\ p_{3} & q_{3} & r_{3}\end{bmatrix}$$ where $$p_{1}, q_{1}, r_{1}$$ are the co-factors of the elements $$l_{i},m_{i},n_{i}$$ for $$i = 1, 2, 3$$. If $$(l_{1}, m_{1}, n_{1}), (l_{2}, m_{2}, n_{2})$$ and $$(l_{3},m_{3}, n_{3})$$ are the direction cosines of three mutually perpendicular lines then $$(p_{1}, q_{1}, r_{1}), (p_{2}, q_{2}, r_{2})$$ and $$(p_{3}, q_{3}, r_{3})$$ are

If a lines makes angles $$ \alpha , \beta , \gamma , \delta $$ with four digonals of a cube. Then $$ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma + \cos ^2 \delta $$ will be :

ABC is a triangle where $$A(2,3,5), B(-1,3,2)$$ and $$C(\lambda , 5, \mu)$$. Let the median through $$A$$ is equally inclined to the axes.
The value of $$\mu - \lambda$$ is equal to:

The point of intersection of the line joining the points $$(-3, 4, -8)$$ and $$(5, -6, 4)$$ with the $$XY$$-plane is

Find the direction cosines of the line which is perpendicular to the lines with direction cosines proportional to 1, -2, -2 and 0, 2, 1.