Three Dimension...

The vector equation of the plane through the point $$\vec i+2\vec j-\vec k$$ and $$\bot$$ to the line of intersection of the plane $$\overrightarrow { r } .\left( 3\vec i-\vec j+\vec k \right) =1$$ and $$\overrightarrow { r } .\left(\vec  i+4\vec j-2\vec k \right) =2$$ is

The direction cosine of a line equally inclined to the axes are

Equation of the line which passes through the point with p.v. (2, 1, 0) and perpendicular to the plane containing the vectors $$\widehat{i}+\widehat{j}\:and\: \widehat{j}+\widehat{k}$$ is

Statement-1  :  If a line makes acute angles $$\alpha, \beta, \gamma, \delta$$ with diagonals of a cube, then $$ \displaystyle \cos^2\alpha+\cos^2\beta+\cos^2\gamma+\cos^2\delta=\frac{4}{3}$$
Statement 2  :  If a line makes equal angle (acute) with the axes, then its direction cosine are $$ \displaystyle \frac{1}{\sqrt{3}} , \frac{1}{\sqrt{2}}$$ and $$\dfrac{1}{\sqrt{3}}$$

The direction cosines of a line whose equations are $$\dfrac{x-1}{2}=\dfrac{y+3}{4}=\dfrac{z-2}{-3}$$

If the foots of the perpendicular from the origin to a plane is $$(a,b,c)$$, the equation of the plane is

If equation of the plane through the straight line $$\displaystyle \dfrac{x -1}{2}=\dfrac{y +2}{-3}=\dfrac{z}{5}$$ and perpendicular to the plane $$x - y + z + 2 = 0 \: $$is$$ \:ax- by + cz + 4 = 0,$$ then find the value of  $$ a^2 + b^2 + c$$

The line $$\displaystyle \frac{x - 1}{2} = \frac{y}{-1} = \frac{z + 2}{2}$$ cuts the plane $$\displaystyle x + y + z = 1$$ at $$\displaystyle P$$. If the foot of the perpendicular from $$\displaystyle P$$ to a point $$Q\displaystyle \left ( 3, \: -4, \: 1 \right )$$ on the plane $$S$$ then the equation of the plane $$S$$ is

if a line makes angles $$\alpha,\beta,\gamma,\delta$$ with four diagonals a cube then value of $$sin^{2}\alpha+sin^{2}\beta+sin^{2}\gamma+sin^{2}\delta$$ equals 

Determine the equation of the plane on which the co - ordinates of the foot of perpendicular drawn from origin O is the point $$\displaystyle P\left ( \alpha ,\beta ,\gamma  \right ).$$