Three Dimension...

Let the direction - cosines of the line which is equally inclined to the axis be $$\displaystyle \pm \frac{1}{\sqrt{k}}$$. Find $$k$$ ?

If a line makes an angle $$\displaystyle \theta_1, \theta_2, \theta_3$$ which the axis respectively, then $$\displaystyle cos 2\theta_1 + cos 2 \theta_2 + cos 2 \theta_3 = ?$$

The direction ratios of a normal to the plane through $$\left( 1,0,0 \right) ,\left( 0,1,0 \right) ,$$ which makes an angle of $$\displaystyle \frac { \pi  }{ 4 } $$ with the plane $$x+y=3$$ are 

let $$P(4,1,\lambda)$$ and $$Q(2,-1,\lambda)$$ be two points. A line having direction ratios $$1,-1,6$$ is perpendicular to the plane passing through the origin, $$P$$ and $$Q$$, then $$\lambda$$ equals

$$\bar a,\bar b,\bar c$$ are three non-zero vectors such that any two of them are non-collinear. If  $$\bar a+\bar b$$ is collinear with  $$\bar c$$ and  $$\bar b+\bar c$$ is collinear with $$\bar a$$, then what is their sum?

The equation of plane through $$(1, 2, 3)$$ and parallel to the plane $$\bar{r}.\left ( \hat{3i}+\hat{4j}+\hat{5k} \right )=0$$

The acute angle between two lines whose direction ratios are $$2,3,6$$ and $$1,2,2$$ is

Equation of the plane which passes through the point (-1, 3, 2) & is $$ \perp  $$ to each of the planes $$ p_{1} $$ & $$ p_{2} $$ is

Equation of the plane passing through a point with position vector $$\displaystyle  3\hat{i}-3\hat{j}+\hat{k} $$ & normal to the line joining the points with position vectors $$\displaystyle  3\hat{i}+4\hat{j}-\hat{k} $$ & $$\displaystyle  2\hat{i}-\hat{j}=5\hat{k} $$ is

If a point $$P$$ in the space such that $$\overline{OP}$$ is inclined to $$OX$$ at $$45$$ and $$OZ$$ to $$60$$ then $$\overline{OP}$$ inclined to $$OY$$ is