Practice Questions

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.

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

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

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

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

Define skew lines.

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.

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

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

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

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

Define direction cosines of a directed line in space.

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.

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.

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

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.

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 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 $\vec{b} = a\hat{i} + b\hat{j} + c\hat{k}$.

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

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

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.

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 $L_2: \frac{x+2}{-1} = \frac{y-4}{8} = \frac{z-5}{4}$.

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 $(3, 4, -2)$.

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 $\vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(2\hat{i} + \hat{j} + 2\hat{k})$.

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.

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.

Write the vector equation of the z-axis.

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

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

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.

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 $L_2: \vec{r} = (2\hat{i} + 4\hat{j} + 5\hat{k}) + \mu(2\hat{i} + 3\hat{j} + 4\hat{k})$.

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.

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 $(\delta\theta)^2 = (\delta l)^2 + (\delta m)^2 + (\delta n)^2$.

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 $\vec{r} = (2\hat{i} - \hat{j} - 3\hat{k}) + \mu(\hat{i} + 2\hat{j} + 2\hat{k})$.

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.

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.

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

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.