Direction Cosines and Direction Ratios of a Line

Let $P\left(-2,-1,1\right)$ and $Q\left(\frac{56}{17}, \frac{43}{17}, \frac{111}{17}\right)$ be the vertices of the rhombus $PQRS.$ If the direction ratios of the diagonal $RS$ are $\alpha ,-1,\beta$, where both $\alpha \text{and} \beta$ are integers of minimum absolute values, then ${\alpha }^2+{\beta }^2$ is equal to $$

If direction ratios of a line are $\lambda , \sqrt{{\lambda }^2-3},1$ and it makes and angle of ${60}^{\circ }$ with $z-$axis, then a value of $y$ is $?$

Anuj and Tara are flying kites from their rooftops. Anuj's kite's string is represented by a straight line, given by $\frac{x-4}{1}=\frac{y-2}{3}=\frac{z-1}{2}.$ Tara's position is at $\left(2\hat{i}-2\hat{j}+\hat{k}\right),$ and her kite's string is perpendicular to Anuj's. Find out how far Tara is from where the kite strings intersect. Show your work.

A line makes an angle of 135° with the positive direction of the x-axis, and an angle of 300° with the positive direction of the y-axis.   Which of the following could be the angle it makes with the negative direction of the z axis? 

Write the direction cosines of the line, whose Cartesian equations are $6x-12=3y+9=2z-2.$

If $\overrightarrow{a}=\frac{1}{\sqrt{4}}\left(2\hat{i}+3\hat{j}+\hat{k}\right)$ then write the direction cosines of $\overrightarrow{a}$ ?

The direction ratios of a normal to the plane passing through $(0,1,1),(1,1,2)$ and $(-1,2,-2)$ are

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 $\left(2,3,c\right)$ are the direction ratios of a ray passing through the point $C\left(5,q,1\right)$ and also the mid point of the line segment joining the points $A\left(p,-4.2\right)$ and $B\left(3,2,-4\right)$ then $c.\left(p+7q\right)=$

If $\left(a,b,c\right)$ are the direction ratios of a line joining the points $\left(4,3,-5\right)$ and $\left(-2,1,-8\right)$ then the point $P\left(a,3 b,2c\right)$ lies on the plane

If the direction ratios of a line are $1,-3,2$, then the direction cosines of the line are

If the direction ratios $a, b, c$ of a line $L$ satisfy the relation $ab+bc+ca=0$ and $6ab+9bc+8ca=0$, then the direction cosines of the line $L$ are

The point of intersection of the line passing through$\left(0,0,1\right)$ and intersecting the lines $x+2y+z=1$,$-x+y-2z=2$ and $x+y=2, x+z=2$ with $xy$ plane is :

A particle is at a point $A(1,0,1)$ and then moves to the point $B(-1,1,1)$ then angle made by $\overrightarrow{AB}$ with positive direction of $\text{x-axis}$ is

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

Show that the points $A\left(2, 3,-4\right), B\left(1,-2, 3\right)$ and $C\left(3, 8,-11\right)$ are collinear.

Find the direction cosines of $x, y$ and $z$-axis.

Find the direction cosines of the line passing through the two points $\left(-2, 4, -5\right)$ and $\left(1, 2, 3\right)$

If a line makes angle $90°, 60°$ and $30°$ with the positive direction of $x, y$ and $z$-axis respectively, find its direction cosines.

Find the direction cosines of the following line: 

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