3d Geometry

Let $\mathrm{P}(3,2,3), \mathrm{Q}(4,6,2)$ and $\mathrm{R}(7,3,2)$ be the vertices of $\triangle \mathrm{PQR}$. Then, the angle $\angle \mathrm{QPR}$ is

The two lines $x=ay+b,z=cy+d$ and $x = a'y + b',z = c'y + d'$ will be perpendicular, if and only if :

A line makes the same angle $\theta$, with each of the $x$ and $z$ axis. If the angle $\beta \,$, which it makes with y-axis, is such that $\,{\sin ^2}\beta = 3{\sin ^2}\theta ,$ then ${\cos ^2}\theta$ equals :

If a line makes an angle of $\pi /4$ with the positive directions of each of $x$-axis and $y$-axis, then the angle that the line makes with the positive direction of the $z$-axis is :

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

If an angle between the line, ${{x + 1} \over 2} = {{y - 2} \over 1} = {{z - 3} \over { - 2}}$ and the plane, $x - 2y - kz = 3$ is ${\cos ^{ - 1}}\left( {{{2\sqrt 2 } \over 3}} \right),$ then a value of k is :

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

A line $AB$ in three-dimensional space makes angles ${45^ \circ }$ and ${120^ \circ }$ 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 :

ABC is a triangle in a plane with vertices A(2, 3, 5), B(−1, 3, 2) and C($\lambda$, 5, $\mu$). If the median through A is equally inclined to the coordinate axes, then the value of ($\lambda$³ + $\mu$³ + 5) is :

An angle between the lines whose direction cosines are gien by the equations, $l$ + 3m + 5n = 0 and 5$l$m $-$ 2mn + 6n$l$ = 0, is :

If a point R(4, y, z) lies on the line segment joining the points P(2, –3, 4) and Q(8, 0, 10), then the distance of R from the origin is :

Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations l + m $-$ n = 0 and l² + m² $-$ n² = 0. Then the value of sin⁴$\alpha$ + cos⁴$\alpha$ is :

The angle between the straight lines, whose direction cosines are given by the equations 2l + 2m $-$ n = 0 and mn + nl + lm = 0, is :

If two straight lines whose direction cosines are given by the relations $l + m - n = 0$, $3{l^2} + {m^2} + cnl = 0$ are parallel, then the positive value of c is :

Consider a $\triangle A B C$ where $A(1,3,2), B(-2,8,0)$ and $C(3,6,7)$. If the angle bisector of $\angle B A C$ meets the line $B C$ at $D$, then the length of the projection of the vector $\overrightarrow{A D}$ on the vector $\overrightarrow{A C}$ is :

Let $P(x, y, z)$ be a point in the first octant, whose projection in the $x y$-plane is the point $Q$. Let $O P=\gamma$; the angle between $O Q$ and the positive $x$-axis be $\theta$; and the angle between $O P$ and the positive $z$-axis be $\phi$, where $O$ is the origin. Then the distance of $P$ from the $x$-axis is

If the line $\frac{2-x}{3}=\frac{3 y-2}{4 \lambda+1}=4-z$ makes a right angle with the line $\frac{x+3}{3 \mu}=\frac{1-2 y}{6}=\frac{5-z}{7}$, then $4 \lambda+9 \mu$ is equal to :

A plane which passes through the point $(3,2,0)$ and the line ${{x - 4} \over 1} = {{y - 7} \over 5} = {{z - 4} \over 4}$ is :