Three-Dimensional Geometry

Direction ratios of a line are also known as:

If the direction cosines of a line are $(\frac{1}{c}, \frac{1}{c}, \frac{1}{c})$, find the positive value of $c$.

The sum of the squares of the direction ratios $a, b, c$ of a line is always equal to 1. Is this statement true or false?

What are the direction cosines of a line parallel to the $x$-axis?

In the rectangular box shown, the coordinates of the vertex $P$ are $(2, 3, 6)$. What are the direction cosines of the vector representing the diagonal $OP$ starting from the origin $O(0,0,0)$?

A line passes through the origin and makes equal angles with the positive directions of the $x$, $y$, and $z$ axes as shown in the coordinate system. If the line has direction cosines $(l, m, n)$, what is the value of $l^2 + m^2 + n^2$?

What is the projection of a segment of length $R$ on a line with direction cosines $(l, m, n)$ if the segment is parallel to the $x$-axis?

A drone is flying in a straight path from a monitoring station at the origin $O(0, 0, 0)$ to a signal tower located at $P(4, -12, 6)$. A navigation engineer needs to determine the direction cosines of the drone's path to calibrate the satellite tracking system. If the line $OP$ makes angles $\alpha$, $\beta$, and $\gamma$ with the positive $X$, $Y$, and $Z$ axes respectively, find the value of $\cos \alpha + \cos \beta + \cos \gamma$.

If a line has direction ratios proportional to $(a, b, c)$, and it passes through $P(x_1, y_1, z_1)$ and $Q(2, 3, 5)$, where the distance $PQ = 7$ units and the direction cosines are $(\frac{2}{7}, \frac{3}{7}, \frac{6}{7})$, find the point $P$ given $Q$ is further from the origin than $P$ along the line.

A line makes angles $\alpha, \beta, \gamma$ with the coordinate axes. If $\alpha = 45^{\circ}$ and $\beta = 60^{\circ}$, then find the value of $\sin^2 \gamma$.

A line passes through the point $P(2, -3, 4)$ and is parallel to the vector $\vec{b} = 3\hat{i} + 2\hat{j} - 5\hat{k}$. Which of the following represents the Cartesian equation of this line?

Find the vector equation of a line passing through $(2, 3, 2)$ and parallel to the vector $3\hat{i} + 2\hat{j} - 8\hat{k}$.

Two lines with direction ratios $(a_1, b_1, c_1)$ and $(a_2, b_2, c_2)$ are parallel if:

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

If the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $\frac{x-4}{5} = \frac{y-1}{2} = z$ intersect, what is the point of intersection?

The Cartesian equations of a line are $3x+1 = 6y-2 = 1-z$. What is the direction vector of the line?

A line passes through $(2, 1, 3)$ and its direction ratios are $(1, 1, -2)$. Find the coordinates of a point on this line at a distance of $3\sqrt{6}$ from $(2, 1, 3)$.

The image of the point $P(1, 6, 3)$ in the line $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$ is the point $P'$. Calculate the coordinates of $P'$.

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

A line $L$ passes through the point $(1, 1, 1)$ and is parallel to the line of intersection of the planes $x + 2y + z - 1 = 0$ and $x - y + 2z - 7 = 0$. Find the direction ratios of line $L$.

If the shortest distance between two lines is a non-zero value, and the direction vectors are not proportional, the lines are:

The lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$ are parallel. The distance $d$ is given by $d = \frac{|(\vec{a}_2 - \vec{a}_1) \times \vec{b}|}{|\vec{b}|}$. What does $\vec{b}$ represent?

Which of the following describes the shortest distance between two skew lines?

If lines $\vec{r} = \vec{a}_1 + t\vec{b}_1$ and $\vec{r} = \vec{a}_2 + s\vec{b}_2$ are such that $\vec{b}_1 = 2\vec{i} - \vec{j} + \vec{k}$ and $\vec{b}_2 = 4\vec{i} - 2\vec{j} + 2\vec{k}$, the lines are:

Find the shortest distance between the two skew lines $L_1$ and $L_2$ represented in the vector form as follows: $L_1: \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - 3\hat{j} + 2\hat{k})$ $L_2: \vec{r} = (4\hat{i} + 5\hat{j} + 6\hat{k}) + \mu(2\hat{i} + 3\hat{j} + \hat{k})$

Find the shortest distance between the lines $\vec{r} = (\vec{i} + 2\vec{j} - 4\vec{k}) + \lambda(2\vec{i} + 3\vec{j} + 6\vec{k})$ and $\vec{r} = (3\vec{i} + 3\vec{j} - 5\vec{k}) + \mu(2\vec{i} + 3\vec{j} + 6\vec{k})$.

If the lines $\frac{x-1}{k} = \frac{y-2}{2} = \frac{z-3}{3}$ and $\frac{x-2}{3} = \frac{y-3}{k} = \frac{z-1}{2}$ intersect, find the value of $k$.

Two non-intersecting pipes are laid in a factory. Pipe 1 lies along the line $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and Pipe 2 lies along the line $\frac{x-2}{4} = \frac{y-4}{6} = \frac{z-5}{8}$. Determine the distance between these two parallel pipes.

Calculate the shortest distance between the lines $L_1$ and $L_2$ where $L_1$ passes through $(6, 2, 2)$ with direction ratios $(1, -2, 2)$ and $L_2$ passes through $(-4, 0, -1)$ with direction ratios $(3, -2, -2)$.

The equations of two paths are given by $\vec{r} = (\hat{i} + 2\hat{j} - 4\hat{k}) + t(2\hat{i} + 3\hat{j} + 6\hat{k})$ and $\vec{r} = (3\hat{i} + 3\hat{j} - 5\hat{k}) + s(2\hat{i} + 3\hat{j} + 6\hat{k})$. Since these lines are parallel, calculate the shortest distance between them.

What is the intercept of the plane $\vec{r} \cdot (2\hat{i} - \hat{j} + \hat{k}) = 4$ on the $y$-axis?

Find the equation of the plane passing through the point $(1, -1, 2)$ and parallel to the plane $3x + 4y - 5z = 0$.

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

Find the vector equation of the plane whose Cartesian equation is $3x - 4y + 5z = 7$.

A plane $\pi$ passes through the point $P(2, 3, 1)$ and is perpendicular to the vector $\vec{n} = 3\hat{i} - \hat{j} + 2\hat{k}$ as shown in the coordinate representation. If the Cartesian equation of this plane is given by $ax + by + cz = d$, where $a, b, c$ are the components of $\vec{n}$, find the value of $d$.

Find the point of intersection of the line $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and the plane $x + y + z = 15$.

The vector equation of a plane which is at a distance of 7 units from the origin and has $\vec{n} = 3\hat{i} + 5\hat{j} - 6\hat{k}$ as a normal vector is:

A particle moves from point $A(1, 2, 1)$ to point $B(3, 4, -1)$. A barrier is represented by the plane $2x - y + z = 5$. Determine the ratio in which this plane divides the line segment $AB$.

Find the equation of the plane passing through the points $(1, -1, 2)$ and $(2, -2, 2)$ and which is perpendicular to the plane $6x - 2y + 2z = 9$.

A specialized piece of laboratory equipment is positioned such that its base lies on a plane with intercepts $a, b, c$ on the coordinate axes. If the distance from the origin to this plane is $p$, which of the following relationships must hold?

The angle between the line $\frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{2}$ and the $z$-axis is:

What is the angle between the planes $x+y+z=1$ and $x-y+z=1$?

The angle between the lines $\frac{x-5}{7} = \frac{y+2}{-5} = \frac{z}{1}$ and $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ is:

Find the angle between the line $\vec{r} = (1, 2, -1) + \lambda(1, -1, 1)$ and the plane $\vec{r} \cdot (2, 1, 1) = 4$.

Find the angle between the line passing through the points $A(3, -4, -2)$ and $B(12, 2, 0)$ and the plane $3x - y + z = 1$.

Find the value of $\lambda$ so that the line $\frac{x-1}{-3} = \frac{y-2}{2\lambda} = \frac{z-3}{2}$ and $\frac{x-1}{3\lambda} = \frac{y-1}{1} = \frac{z-6}{-5}$ are perpendicular.

The angle between the line $\frac{x+1}{2} = \frac{y}{3} = \frac{z-3}{6}$ and the plane $10x + 2y - 11z = 3$ is:

A laser beam is projected from the origin $(0, 0, 0)$ towards a reflector located at point $P(2, 2, 1)$. This beam represents a line $L$. A protective flat shield is placed in the workspace, defined by the equation $3x - 4y + 5z = 10$. A safety engineer needs to calculate the acute angle $\theta$ between the path of the laser beam and the protective shield to ensure the reflection doesn't hit sensitive equipment. What is the value of $\sin \theta$?

A specialized sensor is placed at the intersection of the planes $x+y+z=1$ and $x-2y+3z=2$. The sensor measures the angle between these two planes. What is the measure of this angle?

Consider two lines $L_1$ and $L_2$ with direction cosines $(l_1, m_1, n_1)$ and $(l_2, m_2, n_2)$. If the lines are inclined to each other at an angle $\theta = 60^\circ$ and $l_1l_2 + m_1m_2 + n_1n_2 = k$, find the value of $k$.

The distance of the point $(2, 3, 4)$ from the plane $3x - 6y + 2z + 11 = 0$ is:

Calculate the distance of point $(2, 1, 0)$ from the plane $2x + y + 2z + 5 = 0$.

The distance of the point $(1, 2, 3)$ from the plane $x - y + z = 5$ is:

What is the distance of the point $(7, -3, -4)$ from the plane $z = 0$?

If the length of the perpendicular from the origin to the plane $x + 2y - 2z + d = 0$ is $4$ units, and $d > 0$, determine the value of $d$.

Find the perpendicular distance from the point $P(2, -1, 5)$ to the plane represented by the equation $2x - 3y + 6z + 7 = 0$.

Find the ratio in which the plane $2x + 3y + 5z = 1$ divides the line segment joining $(1, 2, 1)$ and $(3, 4, 8)$.

Find the equation of the plane which is at a distance of $3\sqrt{3}$ units from the origin and whose normal is equally inclined to the coordinate axes.

A tetrahedron has vertices at $O(0,0,0)$, $A(1,2,1)$, $B(2,1,3)$, and $C(-1,1,2)$. Find the length of the altitude from vertex $C$ to the face $OAB$.

A light source is at $L(1, 1, 1)$. It reflects off a mirror surface $x + y + z = 6$. The image of the light source is at $L'$. What is the distance between the light source $L$ and its image $L'$?