Direction Cosines and Direction Ratios of a Vector

In a three-dimensional co-ordinate system, $P,Q$ and $R$ are images of a point $A\left(a,b,c\right)$ in the $xy, yz$ and $zx$ planes respectively. If $G$ is the centroid of triangle $PQR$, then the area of the triangle $AOG$ is ($O$ is the origin)

$A=\left[\begin{array}{ccc}{l}_1& m_1& n_1\\ l_2& m_2& n_2\\ l_3& m_3& n_3\end{array}\right]$ and $B=\left[\begin{array}{ccc}{p}_1& q_1& r_1\\ p_2& q_2& r_2\\ p_3& q_3& r_3\end{array}\right]$ $p_i, q_i, r_i$ are the cofactors of the elements $l_i, m_i, n_i$ for $i=1,2,3.$ If $\left(l_1,m_1,n_1\right),\left(l_2,m_2,n_2\right)$ and $\left(l_3,m_3,n_3\right)$ are the direction cosines of three mutually perpendicular lines, then $\left(p_1,q_1,r_1\right),\left(p_2,q_2,r_2\right)$ and $\left(p_3,q_3,r_3\right)$ are

A line $AB$ in three- dimensional space makes angle $45°$ and $120°$ with the positive $X$-axis and the positive $Y$-axis, respectively. If $AB$ makes an acute angle $\theta$ with the positive $Z$-axis, then $\theta$ equals

The angle between the lines whose direction cosines satisfy the equations $l+m+n=0$ and $l^2=m^2+n^2$ is

The direction ratios of line $l_1$ passing through $P\left(1,3,4\right)$ and perpendicular to line $l_2=\frac{x–1}{2}=\frac{y–2}{3}=\frac{z–3}{4}$ (where $l_1$ and $l_2$ are coplanar) is

$l_1,m_1,n_1$ and $l_2,m_2,n_2$ are the direction cosines of the two lines inclined to each other at an angle $\theta$ then the direction cosines of the internal bisector of the angle between these lines are

If $P$ is a point in the space such that $OP$ is inclined to $OX$ at $45°$and $OY$ at $60°$, then $OP$ is inclined to $OZ$ at

A line passes through the points $\left(6,-7,-1\right)$ and $\left(2,-3,1\right)$. The direction cosines of the line so directed that the angle made by it with the positive direction of $x$-axis is acute, are

If the angle $\theta$ the between the line $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{2}$ and the plane $2x-y+\sqrt{\lambda }z+4=0$ is such that $\sin \theta =\frac{1}{3}$. The value of $\lambda$ is

Let $L$ be the line of intersection of the planes $2x+3y+z=1$ and $x+3y+2z=2$. If line makes an angle $\alpha$ with positive $x$ -axis, then $\cos \alpha$ equals

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

From a point $P\left(\lambda ,\lambda ,\lambda \right)$ perpendiculars $PQ \& PR$ are drawn respectively on the lines $y=x, z=1$ and $y=-x, z=-1$. If $P$ is such that $\angle QPR$ is right angle, then the possible value $\left(s\right)$ of $\lambda$ is (are)

Prove that the two lines whose direction cosines are connected by the two relations $al+bm+cn=0$ and $u{l}^2+v{m}^2+w{n}^2=0$ are perpendicular if $a^2\left(v+w\right)+b^2\left(w+u\right)+c^2\left(u+v\right)=0$ and parallel if $\frac{a^2}{u}+\frac{b^2}{v}+\frac{c^2}{w}=0$

Let $PM$ be the perpendicular from the point $P(1,2,3)$ to $XY$ plane. If $\overrightarrow{OP}$ makes an angle $\theta$ with the positive direction of $Z$-axis and $\overrightarrow{OM}$ makes an angle $\varphi$ with the positive direction of $X$-axis, where $O$ is the origin and $\theta$ and $\varphi$ are acute angles, then

Consider the planes $P_1: 2x+y+z+4=0, P_2: y-z+4=0$ and $P_3: 3x+2y+z+8=0.$ Let $L_1,L_2,L_3$ be the lines of intersection of the planes $P_2 \& P_3, P_{3 }\& P_1$, and $P_1 \& P_2$, respectively. Then,

Which of the following statements is/are correct?

Find the direction cosine of line which is perpendicular to the lines with direction ratio $\left[1,–2,–2\right]$ and $\left[0,2,1\right]$.

A line make angle $\alpha , \beta , \gamma , \delta$ with four diagonals of cube, then prove that ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma +{\cos }^2\delta =\frac{4}{3}$

If $\cos \alpha , \cos \beta , \cos \gamma$ are the direction cosine of a line, then find the value of ${\cos }^2\alpha +\left(\cos \beta +\sin \gamma \right)\left(\cos \beta -\sin \gamma \right)$.

If $\alpha ,\beta$ and$\mathrm{\gamma }$are the angles made by a line with positive direction of $X$-axis, $Y$-axis and $Z$-axis respectively, then find the value of

$\cos 2\alpha +\cos 2\beta +\cos 2\gamma .$