Direction Cosines and Direction Ratios

A line is inclined at an angle $\alpha , \beta , \gamma$ with the positive direction of $x, y, z$ axes respectively. If $\beta +\gamma =\frac{\mathrm{\pi }}{2}$, then the maximum possible value of ${\left(\cos \alpha +\sin \beta +\sin \gamma \right)}^2$ is equal to

If a line in the space makes angle $\alpha ,\beta$ and $\gamma$ with the coordinates axes, then $\cos \left(\alpha +\beta \right)\cos \left(\alpha -\beta \right)+\cos \left(\beta +\gamma \right)\cos \left(\beta -\gamma \right)+\cos \left(\gamma +\alpha \right)\cos \left(\gamma -\alpha \right)$ is equal to

Find the sum of direction cosines of the line passing through two points $(4, 2, 3)$ and $(4, 5, 7)$.

The direction cosines of the line $\frac{x-2}{2}=\frac{y+1}{-2}=\frac{z-1}{-1}$ are

The direction ratios of the line which is perpendicular to the lines with direction ratios $1,3,2$ and $-1,1,2$ is.

If a line makes angles $\alpha ,\beta ,\gamma$ with the co-ordinate axes, then the value of $\cos 2\alpha +\cos 2\beta +\cos 2\gamma +1$ is

If a line makes angles $90°,135°,45°$ with $X\mathit{,}Y$ and $Z$ axes respectively, then sum of its direction cosines is

If a variable line in two adjacent positions has direction cosines $l, m, n$ and $l+\delta l, m+\delta m, n+\delta n$, show that the small angle $\delta \theta$ between two positions is given by $(\delta \theta {)}^2=(\delta l{)}^2+(\delta m{)}^2+(\delta n{)}^2$.

If the line passing through origin makes angles ${\mathrm{\theta }}_1, {\mathrm{\theta }}_2, {\mathrm{\theta }}_3$ with the planes $XOY, YOZ$ and $ZOX$ respectively, then prove that ${\cos }^2{\mathrm{\theta }}_1+{\cos }^2{\mathrm{\theta }}_2+{\cos }^2{\mathrm{\theta }}_3=2$.

$ABC$ is a triangle where $A\equiv (2, 3, 5)$, $B\equiv (-1, 3, 2)$ and $C\equiv (\lambda , 5, \mu )$. If the median through $A$ is equally inclined to the axes, then find the values of $\lambda$ and $\mu$.

A line makes angles of measure $\frac{\pi }{6}$ and $\frac{\pi }{3}$ with $X$ and $Z$ axes respectively, the angle made by the line with the $Y-$axis. [Enter the value in degrees excluding degree symbol]

A line passes through the points $\left(3, 1, 2\right)$ and $\left(5, -1, 1\right),$ find the direction ratios and direction cosines of the line.

If a line makes angles $45°,135°$ and $\theta$ with the $X, Y$ and $Z$ -axes respectively, then value of $\theta$ is

Find the direction cosines of the sides of the triangle whose vertices are $\left(3, 5, -4\right), \left(-1, 1, 2\right)$ and $\left(-5, -5, -2\right)$. What type of triangle is it?

A line makes angles $\alpha , \beta , \gamma , \delta$ with the four diagonals of a cube, prove that ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma +{\cos }^2\delta =\frac{4}{3}$.

The direction cosines of two lines are determined by the relation $l-5 m+3n=0$ and $7l^2+5m^2-3n^2=0,$ find them.

If a line makes angles $\alpha , \beta , \gamma$ with coordinate axes, prove that $\cos 2\alpha +\cos 2\beta +\cos 2\gamma =-1$.

If a line makes angles $\alpha , \beta , \gamma$ with coordinate axes, prove that ${\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma =2$.

Can $\frac{2}{\sqrt{3}}, \frac{-2}{\sqrt{3}}, \frac{-1}{\sqrt{3}}$ be the direction ratios of any directed line? Justify your answer.

Find the length of the perpendicular from $(2,-3,1)$ to the line $\frac{x+1}{2}=\frac{y-3}{3}=\frac{z+2}{-1}$.