Three Dimensional Geometry

The direction ratios of two lines are $(2, 3, 6)$ and $(1, 2, 2)$. Find the sum of the products of their corresponding direction ratios.

Find the direction cosines of a line joining the origin to the point $(1, 2, 3)$.

The coordinates of a point $P$ are $(x, y, z)$ and the direction cosines of the line $OP$ are $l, m, n$. If $OP = r$, then $x$ is equal to:

The direction cosines of the $x$-axis are:

Consider the line segment joining points $A(2, 3, 5)$ and $B(-1, 2, 4)$. Calculate the direction cosines of the line directed from $A$ to $B$.

If a line has direction ratios proportional to $2, -1, -2$, determine its direction cosines $(l, m, n)$.

A line $L$ passes through the origin $O$ and makes angles $\alpha$, $\beta$, and $\gamma$ with the positive directions of the $x$, $y$, and $z$ axes respectively, as shown in the coordinate system. If $\cos \alpha = \frac{1}{2}$ and $\cos \beta = \frac{1}{\sqrt{2}}$, find the possible value of $\cos \gamma$.

If a line $L$ is equally inclined to the coordinate axes and passes through the point $(2, 3, 4)$, find the coordinates of a point on the line at a distance of $\sqrt{3}$ units from the given point.

Find the direction ratios of the line which is perpendicular to the lines with direction ratios $(1, -2, -2)$ and $(0, 2, 1)$.

A line makes angles $\alpha, \beta, \gamma$ with the X, Y, and Z axes respectively. If $\alpha = \beta$ and the line is perpendicular to a line with direction ratios $(1, 1, 0)$, what is the value of $\gamma$?

The direction cosines $l, m, n$ of a line satisfy $l^2 + m^2 + n^2 = 1$. If $l=m=n$, find the value of $l$.

What is the distance of the point $(2, 3, 4)$ from the $x$-axis?

Are the lines $\frac{x-5}{7} = \frac{y+2}{-5} = \frac{z}{1}$ and $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ perpendicular?

If a line has direction ratios $(2, -1, -2)$, what are its direction cosines?

A line $L$ passes through the point $P(2, -1, 3)$ and is parallel to the vector $\vec{b} = \hat{i} + 2\hat{j} - 2\hat{k}$. If a point $Q(x, y, z)$ on this line is at a distance of $9$ units from $P$ in the direction of $\vec{b}$, find the Cartesian coordinates of $Q$.

If the Cartesian equations of a line are $x=ay+b, z=cy+d$, find its direction ratios.

Find the Cartesian equation of a line passing through $(1, 2, -4)$ and perpendicular to the lines $\frac{x-8}{3} = \frac{y+19}{-16} = \frac{z-10}{7}$ and $\frac{x-15}{3} = \frac{y-29}{8} = \frac{z-5}{-5}$.

If the lines $\frac{x-1}{-3} = \frac{y-2}{-2y_1} = \frac{z-3}{2}$ and $\frac{x-1}{k} = \frac{y-2}{1} = \frac{z-3}{5}$ are coplanar and intersect, find the value of $k$ if they are also perpendicular.

Determine the value of $k$ so that the lines $\frac{x-1}{-3} = \frac{y-2}{2k} = \frac{z-3}{2}$ and $\frac{x-1}{3k} = \frac{y-1}{1} = \frac{z-6}{-5}$ are perpendicular to each other.

A line passes through the point with position vector $2\hat{i} - \hat{j} + 4\hat{k}$ and is in the direction $\hat{i} + 2\hat{j} - \hat{k}$. Find the shortest distance from the point $(3, 3, -1)$ to this line.

Which of the following is true for two skew lines?

The lines $\frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ and $\frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2}$ are parallel if:

What is the relation between the shortest distance $d$ and the coplanarity of two lines?

If the shortest distance between two lines is zero, the lines must be:

Find the shortest distance between the two lines $L_1$ and $L_2$ whose vector equations are given below: $L_1: \vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$ $L_2: \vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k})$

The shortest distance between the z-axis and the line $x+y+2z-3=0$, $2x+3y+4z-4=0$ is:

If the shortest distance between the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{\lambda}$ and $\frac{x-2}{1} = \frac{y-4}{4} = \frac{z-5}{5}$ is $0$, find $\lambda$.

A robot arm moves along a path $L_1$ passing through $(8, -19, 10)$ with direction $(3, -16, 7)$ and another arm moves along $L_2$ passing through $(15, 29, 5)$ with direction $(3, 8, -5)$. Find the shortest distance between the two arms.

Two satellites move along paths $S_1: \vec{r} = (1-t)\hat{i} + (t-2)\hat{j} + (3-2t)\hat{k}$ and $S_2: \vec{r} = (s)\hat{i} + (2s-1)\hat{j} - (s+1)\hat{k}$. Expressing these in standard vector form $\vec{a} + \lambda \vec{b}$, determine if the paths are coplanar.

Verify if the lines $L_1: \frac{x-1}{3} = \frac{y-1}{-1} = \frac{z-1}{0}$ and $L_2: \frac{x-4}{2} = \frac{y-0}{0} = \frac{z+1}{3}$ are skew or intersecting. If they are skew, find the shortest distance.

What is the reflection of the point $(1, 2, -1)$ in the plane $z = 0$?

The direction cosines of the normal to the plane $x + y + z = 1$ are:

If a plane is parallel to the $yz$-plane, its equation is of the form:

Two planes $a_1x + b_1y + c_1z + d_1 = 0$ and $a_2x + b_2y + c_2z + d_2 = 0$ are parallel if:

A plane $\pi$ passes through the point $P(2, 3, -1)$ and is perpendicular to the vector $\vec{n} = 3\hat{i} - 4\hat{j} + \hat{k}$. If the distance of the origin from this plane is $d$, and the point $Q(k, 1, 1)$ lies on the plane, find the value of $k$.

Find the length of the perpendicular from the point $(2, 3, 4)$ to the plane $3x - 6y + 2z + 11 = 0$.

The Cartesian equation of the line of intersection of the planes $x+y+z=1$ and $x-2y+3z=4$ can be found by setting $z=t$. What is the direction of this line?

The foot of the perpendicular drawn from the origin to a plane is $(12, -4, 3)$. Find the Cartesian equation of this plane.

Find the distance between the parallel planes $\vec{r} \cdot (2\hat{i} - 3\hat{j} + 6\hat{k}) = 5$ and $\vec{r} \cdot (6\hat{i} - 9\hat{j} + 18\hat{k}) + 20 = 0$.

A point $P$ moves such that the sum of the squares of its distances from the planes $x + y + z = 0$, $x - z = 0$ and $x - 2y + z = 0$ is 9. Find the equation of the locus of $P$.

Find the angle between the line $\frac{x}{2} = \frac{y}{2} = \frac{z}{1}$ and the line $\frac{x-5}{4} = \frac{y-2}{1} = \frac{z-3}{8}$.

If a line makes angles $\alpha, \beta, \gamma$ with the coordinate axes, then $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma =$

The direction ratios of two lines are $(1, 1, 0)$ and $(1, 2, 1)$. The angle between them is:

If the angle between the planes $2x - y + z = 11$ and $x + y + 2z = 3$ is $\theta$, then $\theta$ is:

The angle between two planes $x+2y+2z-3=0$ and $3x+4y+5=0$ is:

The angle between the line $\vec{r} = (\hat{i} + \hat{j}) + \lambda(\hat{i} + \hat{k})$ and the plane $\vec{r} \cdot (\hat{i} + \hat{j}) = 5$ is:

The angle between the lines with direction ratios $(a, b, c)$ and $(b-c, c-a, a-b)$ is:

In a crystal lattice, two planes of atoms are given by the equations $2x - y + z = 1$ and $x + y + 2z = 3$. An X-ray is diffracted at an angle. To understand the geometry, calculate the angle between these two lattice planes.

A vertical pole of a tent is represented by the line segment on the line $\frac{x-2}{1} = \frac{y-1}{2} = \frac{z-3}{2}$. If the tent fabric on one side is the plane $2x - y + z = 6$, find the angle between the pole and the tent fabric.

A solar panel is mounted on a roof. The roof is the plane $x + y + z = 1$ and the solar panel's surface is the plane $2x + y - z = 5$. Determine the angle $\theta$ between the solar panel and the roof.

The distance of the point $(3, 4, 5)$ from the plane $2x + y - 2z = 5$ is:

The distance of the point $(1, -1, 1)$ from the plane $\vec{r} \cdot (2\hat{i} - 2\hat{j} + \hat{k}) + 5 = 0$ is:

Find the distance of the point $(2, 3, -5)$ from the plane $x + 2y - 2z = 9$.

Distance of the point $(1, -2, 4)$ from the plane $x - 2y + z + 5 = 0$ is:

Find the perpendicular distance from the point $P(2, 5, -3)$ to the plane given by the equation $6x - 3y + 2z - 4 = 0$.

The point on the x-axis which is at a distance of $\sqrt{14}$ units from the plane $3x - 2y + z + 1 = 0$ is:

Find the vector equation of a plane which is at a distance of 7 units from the origin and has normal vector $3\hat{i} + 5\hat{j} - 6\hat{k}$.

A spherical ball of radius $2$ units is placed such that its center is at $C(3, 4, 5)$. A flat plate is represented by $2x + 2y + z = 1$. Find the minimum clearance distance between the surface of the ball and the plate.

A point $M(a, b, c)$ is at a distance of $d$ from the plane $Ax + By + Cz + D = 0$. If the plane is shifted parallel to itself such that the new distance of $M$ from the plane is $2d$, which of the following could be the equation of the new plane?

Find the distance between the point $Q(2, 3, -1)$ and the plane passing through the points $(3, 4, 2)$, $(7, 0, 6)$, and $(0, 3, 1)$.

Find the direction cosines of the line passing through origin and the point $(1, 1, 0)$.

The direction cosines of a line joining origin to $(a, b, c)$ are proportional to:

If the direction ratios of a line are $(1, 2, 2)$, what are its direction cosines?

Given two points $A(2, 3, 5)$ and $B(4, 6, 11)$, find the direction ratios of line $AB$.

A line passes through $(2, 1, 3)$ and $(-1, 2, 1)$. What is the direction cosine $n$?

Find the direction cosines of a line that is parallel to the line joining $(2, 4, 5)$ and $(1, 2, 3)$.

Points $A(2, 3, -4)$, $B(1, -2, 3)$ and $C(3, 8, -11)$ are collinear. Find the direction ratios of the line passing through them.

A line makes equal angles with the coordinate axes. If the line passes through $(1, 2, 3)$ and $(x, y, z)$, what is a possible set of direction cosines for this line?