Practice Questions

Define direction cosines of a directed line in space.

Define skew lines.

State the relationship that exists between the direction cosines ($l, m, n$) of any line.

Identify the direction ratios of the line given by the Cartesian equation $\frac{x-5}{2} = \frac{y+3}{-4} = \frac{z}{6}$.

Explain the difference between direction cosines and direction ratios of a line.

A line has direction ratios $6, 2, -3$. Calculate its direction cosines.

Describe the condition for two lines with direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ to be (i) perpendicular and (ii) parallel.

State the vector and Cartesian equations for a line passing through two given points. Identify each component in the equations.

Determine the vector equation of a line that passes through the point $(5, -2, 4)$ and is parallel to the vector $2\hat{i} - \hat{j} + 3\hat{k}$.

Find the Cartesian form of the equation of the line given by the vector equation $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - 2\hat{j} + 2\hat{k})$.

Calculate the vector and Cartesian equations for the line passing through the points $A(-1, 0, 2)$ and $B(3, 4, 6)$.

Justify why the direction cosines of a line uniquely determine its direction in space, whereas direction ratios do not.

Formulate a condition, using the dot product, to critique whether two lines, given in vector form $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$, are perpendicular. Justify your formulation.

Write the vector equation of the z-axis.

Evaluate the statement: "The shortest distance between any two lines in space is always found along the line segment that is perpendicular to both lines." Is this statement universally true? Justify your answer.

A line L passes through the point P(1, 2, 3) and is perpendicular to two other lines L1 and L2 with direction ratios $<1, -2, 3>$ and $<2, 1, -1>$ respectively. Formulate the Cartesian equation of the line L.

State the formula for the shortest distance between two skew lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}_1$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}_2$.

Evaluate whether the line joining points A(1, 2, 3) and B(-1, -2, -3) is perpendicular to the line joining points C(4, 1, 5) and D(3, 5, 2). Justify your conclusion.

Examine if the points $A(1, 2, 3)$, $B(4, 0, 4)$, and $C(-5, 6, 1)$ are collinear.

Calculate the shortest distance between the skew lines $l_1: \frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$ and $l_2: \frac{x-2}{3} = \frac{y-4}{4} = \frac{z-5}{5}$.

Explain how to derive the Cartesian form of the equation of a line from its vector form $\vec{r} = \vec{a} + \lambda \vec{b}$. Let $\vec{a} = x_1\hat{i} + y_1\hat{j} + z_1\hat{k}$ and $\vec{b} = a\hat{i} + b\hat{j} + c\hat{k}$.

Summarize the derivation of the formula for the direction cosines of a line passing through two points $\mathrm{P}(x_1, y_1, z_1)$ and $\mathrm{Q}(x_2, y_2, z_2)$.

Describe the concept of the angle between two lines in space. State the formula to find this angle in terms of (i) their direction cosines and (ii) their direction ratios.

If a line makes angles of $90^\circ$ and $60^\circ$ with the positive x and y-axes respectively, find the angle it makes with the positive z-axis.

Calculate the angle between the pair of lines given by the equations: $L_1: \frac{x-2}{2} = \frac{y-1}{5} = \frac{z+3}{-3}$ and $L_2: \frac{x+2}{-1} = \frac{y-4}{8} = \frac{z-5}{4}$.

List the direction cosines of a line that is equally inclined to the positive coordinate axes.

Create the vector equation of a line that passes through the origin and is equally inclined to the positive coordinate axes.

Justify that the points A(1, -1, 3), B(2, -4, 5), and C(5, -13, 11) are collinear. Further, formulate the ratio in which point B divides the line segment AC.

Show that the line passing through the points $(0, 3, 2)$ and $(3, 5, 6)$ is perpendicular to the line passing through the points $(1, -1, 2)$ and $(3, 4, -2)$.

If the direction ratios of a line are $2, -3, 6$, explain the process to find its direction cosines.

Propose a method to determine if three points A, B, and C are collinear using the concept of direction ratios.

Find the value of $p$ so that the lines $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{p}$ and $\frac{x-4}{2} = \frac{y-5}{1} = \frac{z-6}{-3}$ are at right angles.

Calculate the shortest distance between the lines $l_1$ and $l_2$ whose vector equations are $\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k})$.

A line makes angles $45^\circ$, $135^\circ$, and $60^\circ$ with the positive x, y, and z-axes respectively. Recall the formula for direction cosines and find them.

Derive the formula for the shortest distance between two parallel lines $\vec{r} = \vec{a}_1 + \lambda \vec{b}$ and $\vec{r} = \vec{a}_2 + \mu \vec{b}$. Justify each step of your derivation using vector concepts.

A line makes angles $\alpha, \beta, \gamma, \delta$ with the four diagonals of a cube. Prove that $\cos^2\alpha + \cos^2\beta + \cos^2\gamma + \cos^2\delta = \frac{4}{3}$.

Derive the formula for the shortest distance between two skew lines in Cartesian form: $L_1: \frac{x-x_1}{a_1} = \frac{y-y_1}{b_1} = \frac{z-z_1}{c_1}$ and $L_2: \frac{x-x_2}{a_2} = \frac{y-y_2}{b_2} = \frac{z-z_2}{c_2}$. Justify the use of the scalar triple product in your derivation.

Design a comprehensive algorithm to determine the relationship between two lines given in Cartesian form. Your algorithm must be able to classify the lines as intersecting, parallel, skew, or coincident and, where applicable, calculate the point of intersection or the shortest distance. Justify each decision point in your algorithm.

Formulate the equation of a line passing through the point P(2, -1, 3) and perpendicular to the lines $\vec{r} = (\hat{i}+\hat{j}-\hat{k}) + \lambda(2\hat{i}-2\hat{j}+\hat{k})$ and $\vec{r} = (2\hat{i}-\hat{j}-3\hat{k}) + \mu(\hat{i}+2\hat{j}+2\hat{k})$. Then, create a procedure to find the shortest distance between this newly formulated line and the x-axis.

Design a method to find the coordinates of the foot of the perpendicular drawn from a point P($x_1, y_1, z_1$) to the line $\frac{x-a}{l} = \frac{y-b}{m} = \frac{z-c}{n}$. Justify your proposed steps.

Calculate the distance between the parallel lines $L_1: \vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(2\hat{i} + 3\hat{j} + 4\hat{k})$ and $L_2: \vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu(2\hat{i} + 3\hat{j} + 4\hat{k})$.

The vertices of a triangle are $A(1, 2, 3)$, $B(-1, 0, 4)$, and $C(0, 1, 1)$. Find the direction cosines of the median through vertex A.

Find the vector equation of the line passing through the point $(2, -1, 3)$ and perpendicular to the lines $\vec{r} = (\hat{i} + \hat{j} - \hat{k}) + \lambda(2\hat{i} - 2\hat{j} + \hat{k})$ and $\vec{r} = (2\hat{i} - \hat{j} - 3\hat{k}) + \mu(\hat{i} + 2\hat{j} + 2\hat{k})$.

A variable line in two adjacent positions has direction cosines $(l, m, n)$ and $(l+\delta l, m+\delta m, n+\delta n)$. If $\delta\theta$ is the small angle between the two positions, prove that $(\delta\theta)^2 = (\delta l)^2 + (\delta m)^2 + (\delta n)^2$.