Dcs and drs practice · Easy

The distance between two points $P$ and $Q$ is $d$ and the length of their projections of $P Q$ on the co-ordinate planes are $d_1, d_2, d_3$. Then $d_1^2+d_2^2+d_3^2=k d^2$ where ' $k$ ' is

The angle between the lines whose direction cosines are given by $2 l-m+2 n=0, l m+m n+n l=0$ is

If a line makes angle $\alpha, \beta$ and $\gamma$ with the coordinates axes, then

Consider the following 3 lines in space
${L_1}:r = 3i - j + 2k + \lambda (2i + 4j - k)$
${L_2}:r = i + j - 3k + \mu (4i + 2j + 4k)$
${L_3}:r = 3i + 2j - 2k + t(2i + j + 2k)$
Then, which one of the following pair(s) is/are in the same plane?

If direction cosines of two lines are proportional to $(2,3,-6)$ and $(3,-4,5)$, then the acute angle between them is

If the direction cosines of a line are $\left(\frac{1}{c}, \frac{1}{c}, \frac{1}{c}\right)$ then

If the co-ordinates of the points $P$ and $Q$ be (1, 2, 1) and $(2,3,4)$ and $O$ be the origin, then

The plane containing the two lines $\frac{x-3}{1}=\frac{y-2}{4}=\frac{z-1}{5}$ and $\frac{x-2}{1}=\frac{y+3}{-4}=\frac{z+1}{5}$ is $11 x+m y+n z=28$ where-

The vector equations of two lines $L_1$ and $L_2$ are respectively
$r = 17i - 9j + 9k + \lambda (3i + j + 5k)$
and $r = 15i - 8j - k + \mu (4i + 3j)$
I. $L_1$ and $L_2$ are skew lines.

II. $(11,-11,-1)$ is the point of intersection of $L_1$ and $L_2$.

III. $(-11,11,1)$ is the point of intersection of $L_1$ and $L_2$.

IV. $\cos ^{-1}\left(\frac{3}{\sqrt{35}}\right)$ is the acute angle between, $L_1$ and $L_2$.

Then, which of the following is true?

If the distance between line $3(x+3)=2(y-5)=z+1$ and a line passing through $(1,-2,0)$ and having direction ratios $(2,3,6)$ is $K$ units, then $K^2$ is equal to

The projection of a line $P Q$, where $P$ is $(0,-1,3)$ and $Q$ is $(4,5,6)$ on a line whose direction ratios are $1,2,3$ is

The projection of a line segment on the coordinate axes are 2, 3, 6. Then, the length of the line segment is

If the direction cosines of a line are $\left(\frac{1}{c}, \frac{1}{c}, \frac{1}{c}\right)$ then

The projections of a vector on the three coordinate axes are $6,-3,2$ respectively. The direction cosines of the vector are

The symmetric form of the equations of the line $x+y-z=1,2 x-3 y+z=2$ is