Direction Cosines and Direction Ratios of a Line

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

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]

Show that points $\left(2, -1, -1\right), \left(4, -3, 0\right)$ and $\left(0, 1, -2\right)$ are collinear.

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.

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.

Find the values of $\lambda$ for which the points $(6, -1, 2),(8, -7, \lambda )$ and $\left(5, 2, 4\right)$ are collinear.

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.

If a line makes $45°, 90°, 135°$ with $x, y, z-$axis respectively then find the product of its direction cosines.

The equations of a line is 5x-3 = 15y +7 = 3-10z.Write the direction cosines of the line.

Find the direction cosine of the line

$\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}.$

If a line makes angles $90°, 135°, 45°$ with the $X, Y$ and $Z$ axes respectively, then find its direction cosines.

The projection of the line segment on the axes are $-3, 4,-12$ respectively. Find the length and direction cosines of the line segment.

The direction cosines of two lines are expressed with the following given relations, find them

$l-5m+3n=0$ and $7l^2+5m^2-3n^2=0$