3D Geometry Questions (578)

Assuming the plane \(4x - 3y + 7z = 0\) to be horizontal, the direction cosines of the line of greatest slope in the plane \(2x + y - 5z = 0\) are

Consider the line $L$ passing through the points $(1,2,3)$ and $(2,3,5)$. The distance of the point $\left(\dfrac{11}{3},\dfrac{11}{3},\dfrac{19}{3}\right)$ from the line $L$ along the line $\dfrac{3x-11}{2}=\dfrac{3y-11}{1}=\dfrac{3z-19}{2}$ is equal to:

The minimum value of $x^2+y^2+z^2$ if $ax+by+cz=p$ is

If point $A$ lies on $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and point $B$ lies on $\dfrac{x-2}{3}=\dfrac{y-4}{7}=\dfrac{z-6}{6}$, then $\overrightarrow{AB}$ cannot be parallel to

The shortest distance between the lines \(\frac{x - 3}{7} = \frac{y - 8}{-6} = \frac{z - 3}{1}\) and \(\frac{x + 3}{1} = \frac{y + 7}{-2} = \frac{z - 6}{1}\) is

The length of the perpendicular from the point \((2, -1, 4)\) on the straight line \(\frac{x + 3}{10} = \frac{y - 2}{-7} = \frac{z}{1}\) is

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

Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three non-zero vectors satisfying $\vec{a}=\vec{b}\times\vec{c}+2\vec{b}$, where $|\vec{b}|=|\vec{c}|=2$ and $|\vec{a}|\leq4$. The sum of possible values of $|2\vec{a}+\vec{b}+\vec{c}|$ is

The equation of the plane containing line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ and perpendicular to plane $4x+5y-3z-8=0$ is

The perpendicular distance of a corner of a unit cube from a diagonal not passing through it is

Consider the set of eight vectors $V=\{a\hat{i}+b\hat{j}+c\hat{k}: a,b,c\in\{-1,1\}\}$. The number of ways three non-coplanar vectors can be chosen from $V$ equals

Consider lines $L_1:\{3\sqrt3x=4\sin\theta\,y+(2\sqrt{\sin\theta}-3)\,3\sqrt3,\;3\sqrt3z=(2\sqrt{\sin\theta}-3)y+3\sqrt3\sin\theta\}$ and $L_2:\{\sqrt3x=-2\sqrt{\cos\theta}\,y+\sqrt3(3-2\sqrt{\cos\theta}),\;\sqrt3z=(3-2\sqrt{\cos\theta})y+\sqrt3\cos\theta\}$. If $L_1\perp L_2$ then

A ladder of 3m length leans against a wall. The ladder forms a vertical angle of 30° with the wall. The top slides down at 20 cm/s. The bottom slides away at 20 cm/s at time $t$. The average velocity of a person halfway up the ladder for the first $t$ seconds is

Let $Q$ be the cube with vertices $\{(x_1,x_2,x_3)\in\mathbb{R}^3:x_1,x_2,x_3\in\{0,1\}\}$. Let $F$ be the set of all 12 lines containing face diagonals and $S$ be the set of 4 main diagonals. For lines $l_1\in F$ and $l_2\in S$, let $d(l_1,l_2)$ denote shortest distance. Maximum of $d(l_1,l_2)$ is $\lambda$. Find $\lambda^{-2}$.

Let \(A_1, A_2, A_3, A_4\) be the areas of the triangular faces of a tetrahedron, and \(h_1, h_2, h_3, h_4\) be the corresponding altitude of the tetrahedron. If volume of tetrahedron is \(1/6\) cubic units, then find the minimum value of \((A_1 + A_2 + A_3 + A_4)(h_1 + h_2 + h_3 + h_4)\) (in cubic units).

If point $A$ lies on $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ and point $B$ lies on $\dfrac{x-2}{3}=\dfrac{y-4}{7}=\dfrac{z-6}{6}$, then $\overrightarrow{AB}$ cannot be parallel to

The line $(x,y,z)=(2,-3,4)+\lambda(1,2,-3)$ intersects $2x+3y-z=13$ at $P$ and $yz$-plane at $Q$. If $PQ=a\sqrt{b}$, $a,b\in\mathbb{N}$, $a>3$, then $\dfrac{a+b}{3}$ equals

The equation of the plane passing through $(1,-1,2)$ and perpendicular to planes $2x+3y-2z=5$ and $x+2y-3z=8$ is

Let $L_1$, $L_2$ be two distinct lines such that $L_1$ and $L_2$ are not perpendicular. Let $P$ be a plane such that $L_1$ and $L_2$ are not perpendicular to $P$. Let $L_3$ and $L_4$ be projections of $L_1$ and $L_2$ on plane $P$ respectively. If $\theta$ be angle between $L_3$ and $L_4$ then $\theta$ CANNOT be equal to

The distance between the line $\vec{r}=2\hat{i}-2\hat{j}+3\hat{k}+\lambda(\hat{i}-\hat{j}+4\hat{k})$ and the plane $\vec{r}\cdot(\hat{i}+5\hat{j}+\hat{k})=5$ is

Statement-1: The point \(A(3, 1, 6)\) is the mirror image of the point \(B(1, 3, 4)\) in the plane \(x - y + z = 5\).Statement-2: The plane \(x - y + z = 5\) bisects the line segment joining \(A(3, 1, 6)\) and \(B(1, 3, 4)\).

A plane meets coordinate axes at $A$, $B$, $C$ such that centroid of $\triangle ABC$ is at $(p,q,r)$. Equation of plane is

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

If $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+7\hat{j}+2\hat{k}$, $\vec{x}\cdot\vec{a}=0$ and $\vec{x}\cdot\vec{c}=0$ for some non-zero vector $\vec{x}$, then value of $\vec{a}\cdot(\vec{b}\times\vec{c})$ is

The distance of point $P(3,8,2)$ from the line $\dfrac{x-1}{2}=\dfrac{y-3}{4}=\dfrac{z-2}{3}$ measured parallel to plane $3x+2y-2z+15=0$ is

Equation of plane containing the line $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$ at minimum possible distance from $(-1,-3,1)$ is

Let $\vec{p},\vec{q}$ and $\vec{r}$ be three unit vectors satisfying $|\vec{p}-\vec{q}|^2+|\vec{q}-\vec{r}|^2+|\vec{r}-\vec{p}|^2=9$. Then $|2\vec{p}+5\vec{q}+5\vec{r}|$ is equal to

Match List-I with List-II: (A) Line $\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-2k}{2}$ lies in plane $2x-4y+z=3$; (B) Lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z}{1}$ intersect, value of $k$; (C) Plane through $(1,1,1)$ with OA=OB=OC, volume of tetrahedron OABC; (D) Distance from $(-1,5/\sqrt{2},3/\sqrt{2})$ to plane $P$ through $(1,-2,1)$ perpendicular to $2x-2y+z=0$ and $x-y+2z=4$

A perpendicular is drawn from a point on the line \(\dfrac{x-1}{2} = \dfrac{y+1}{-1} = \dfrac{z}{1}\) to the plane \(x + y + z = 3\) such that the foot of the perpendicular \(Q\) also lies on the plane \(x - y + z = 3\). The co-ordinates of \(Q\) are:

Let $\vec{a}$, $\vec{b}$ be two vectors perpendicular to each other with $|\vec{a}|=2$, $|\vec{b}|=3$ and $\vec{c}\times\vec{a}=\vec{b}$. The least value of $|\vec{c}-\vec{a}|$ is

The minimum value of $x^2+y^2+z^2$ if $ax+by+cz=p$ is

If direction cosines of a line \(L\) satisfy \(l = -m - n\) and \(l^2 = m^2 + n^2\), then the angle \(\theta\) that the line makes is such that \(\cos\theta =\):

If the angle between the line \(2(x+1) = y = z + 4\) and the plane \(2x - y + \sqrt{\lambda}z + 4 = 0\) is \(\pi/6\), then the value of \(\lambda\) is

The distance of the point \((1, 0, 2)\) from the point of intersection of the line \(\dfrac{x-2}{3} = \dfrac{y+1}{4} = \dfrac{z-2}{12}\) and the plane \(x - y + z = 16\), is

Two systems of rectangular axes have the same origin. If a plane cuts them at distances \(a, b, c\) and \(a', b', c'\) from the origin, then

Line x = ay + b, z = cy + d and line x = a'z + b', y = c' + d' are perpendicular to each other. Then which condition holds?

Given two planes \(P_1: 2x - y - 4 = 0\) and \(P_2: y + 2z - 4 = 0\) and point \(K(1, 1, 0)\). Let a third plane \(P_3\) pass through \(K\) and satisfy \(P_3: P_1 + \lambda P_2 = 0\). The equation of plane \(P_3\) is:

ABC is a triangle with vertices A(0, 0, 6), B(0, 4, 0) and C(6, 0, 0). Let points D, E and F are the mid-points of BC, AC and AB, respectively. Find the length of median AD.

Two lines \(\frac{x - 3}{1} = \frac{y + 1}{3} = \frac{z - 6}{-1}\) and \(\frac{x + 5}{7} = \frac{y - 2}{-6} = \frac{z - 3}{4}\) intersect at the point R. The reflection of R in the xy-plane has coordinates

Let (l, 2, 1) be a point on the plane which passes through the point (4, -2, 2). If the plane is perpendicular to the line joining the points (-2, -21, 29) and (-1, -16, 23), then \(\left(\frac{l}{11}\right)^2 - \frac{4l}{11} - 4\) is equal to

The equation of the plane with intercepts 2, 3 and 4 on the X, Y and Z-axes respectively, is

The equation of the plane passing through the point (0, 7, -7) and containing the line \(\frac{x+1}{-3} = \frac{y-3}{2} = \frac{z+2}{1}\) is

The direction ratios of normal to the plane through the points (0, −1, 0) and (0, 0, 1) and making an angle π/4 with the plane y − z + 5 = 0 are

If the mirror image of the point $P(3,4,9)$ in the line $\dfrac{x-1}{3}=\dfrac{y+1}{2}=\dfrac{z-2}{1}$ is $(\alpha,\beta,\gamma)$, then $14(\alpha+\beta+\gamma)$ is:

Let $(\alpha,\beta,\gamma)$ be the mirror image of the point $(2,3,5)$ in the line $\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}$. Then $2\alpha+3\beta+4\gamma$ is equal to

If the planes $x = cy + bz, y = az + cx, z = bx + ay$ pass through one line then the value of $a^2 + b^2 + c^2 + 2abc$ is __________.

A variable plane is at a constant distance $p$ from the origin and meets the axes at $A, B, C$. If the locus of the centroid of the tetrahedron $OABC$ is $x^{-2} + y^{-2} + z^{-2} = 2qp^{-2}$ then the value of $\sqrt{\lambda}$ is __________.

If the planes $x - y + z + 1 = 0, \lambda x + 3y + 2z - 3 = 0, 3x + \lambda y + z - 2 = 0$ form a triangular prism then $\lambda$ is __________.

If the distance between the plane $Ax - 2y + z = d$ and the plane containing the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-2}{3} = \frac{y-3}{4} = \frac{z-4}{5}$ is $\sqrt{6}$, then $|d|$ is equal to __________.

Let \(A(2, 3, 5)\), \(B(-1, 3, 2)\) and \(C(\lambda, 5, \mu)\) be the vertices of a \(\triangle ABC\). If the median through \(A\) is equally inclined to the coordinate axes, then