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Mathematics · Class 12 Science

Ch 11Three-Dimensional Geometry — Class 12 Mathematics, concept-first.

The study of geometry in three dimensions began in Class XI, where we used Cartesian coordinates to locate points in space. That approach, while powerful, often made the algebra heavy and the geometry less transparent. Now, with the vector algebra developed in the previous chapter, we have a more elegant tool.

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Key concepts

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Direction Cosines Properties

To describe which way a line points in 3D — ignoring its length — we give the angles it makes with the three coordinate axes. Call them (with the x-, y-, z-axis). Their cosines

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

11.1

Introduction

The study of geometry in three dimensions began in Class XI, where we used Cartesian coordinates to locate points in space.

11.2

Direction Cosines and Direction Ratios of a Line

When a directed line passes through the origin and makes angles , , and with the positive , , and -axes respectively, these angles are the direction angles of the line.

11.2.1

Direction Cosines of a Line Passing Through Two Points

10 Q

Through two distinct points in space passes exactly one line, and we can find its direction cosines directly from the coordinates of the two points.

+Worked Examplesi5 questions
  1. Example 1If a line makes angle $90^\circ$, $60^\circ$ and $30^\circ$ with the positive direction of $x$, $y$ and $z$-axis respectively, find its dire…Free
  2. Example 2If a line has direction ratios $2, -1, -2$, determine its direction cosines.Free
  3. Example 3Find the direction cosines of the line passing through the two points $(-2, 4, -5)$ and $(1, 2, 3)$.Preview
  4. Example 4Find the direction cosines of $x$, $y$ and $z$-axis.Preview
  5. Example 5Show that the points $A(2, 3, -4)$, $B(1, -2, 3)$ and $C(3, 8, -11)$ are collinear.Preview
+Exercise 11.1i5 questions
  1. Q1If a line makes angles $90^\circ$, $135^\circ$, $45^\circ$ with the $x, y$ and $z$-axes respectively, find its direction cosines.Free
  2. Q2Find the direction cosines of a line which makes equal angles with the coordinate axes.Free
  3. Q3If a line has the direction ratios $-18, 12, -4$, then what are its direction cosines ?Preview
  4. Q4Show that the points $(2, 3, 4)$, $(-1, -2, 1)$, $(5, 8, 7)$ are collinear.Preview
  5. Q5Find the direction cosines of the sides of the triangle whose vertices are $(3, 5, -4)$, $(-1, 1, 2)$ and $(-5, -5, -2)$.Preview
11.3

Equation of a Line in Space

In two-dimensional geometry, a line is determined by a point and a slope, or by two points. In three-dimensional space, a line is uniquely determined if we know either:

11.3.1

Equation of a Line Through a Given Point and Parallel to Given Vector

The fundamental problem is to find the equation of a line when we know one point it passes through and its direction.

+Worked Examplesi1 question
  1. Example 6Find the vector and the Cartesian equations of the line through the point $(5, 2, -4)$ and which is parallel to the vector $3\hat{i} + 2\hat…Preview
11.4

Angle Between Two Lines

When two lines lie in space, they may intersect, be parallel, or be skew. The angle between them is the acute angle between their direction vectors.

+Worked Examplesi2 questions
  1. Example 7Find the angle between the pair of lines given by $\vec{r} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$ and $\…Free
  2. Example 8Find the angle between the pair of lines $\dfrac{x+3}{3} = \dfrac{y-1}{5} = \dfrac{z+3}{4}$ and $\dfrac{x+1}{1} = \dfrac{y-4}{1} = \dfrac{z-…Preview
11.5

Shortest Distance Between Two Lines

The shortest distance between two lines in space is the length of the smallest possible segment joining a point on one line to a point on the other.

11.5.1

Distance Between Two Skew Lines

Two lines in space that are neither parallel nor intersecting are called skew lines — they do not lie in the same plane and never meet.

11.5.2

Distance Between Parallel Lines

17 Q

When two lines in space are parallel, the shortest distance between them is the length of the perpendicular segment connecting a point on one line to the other.

+Worked Examplesi2 questions
  1. Example 9Find the shortest distance between the lines $l_1$ and $l_2$ whose vector equations are $\vec{r} = \hat{i} + \hat{j} + \lambda(2\hat{i} - \h…Free
  2. Example 10Find the distance between the lines $l_1$ and $l_2$ given by $\vec{r} = \hat{i} + 2\hat{j} - 4\hat{k} + \lambda(2\hat{i} + 3\hat{j} + 6\hat{…Preview
+Exercise 11.2i15 questions
  1. Q1Show that the three lines with direction cosines $\frac{12}{13}, \frac{-3}{13}, \frac{-4}{13}$; $\frac{4}{13}, \frac{12}{13}, \frac{3}{13}$;…Free
  2. Q2Show that the line through the points $(1, -1, 2), (3, 4, -2)$ is perpendicular to the line through the points $(0, 3, 2)$ and $(3, 5, 6)$.Free
  3. Q3Show that the line through the points $(4, 7, 8), (2, 3, 4)$ is parallel to the line through the points $(-1, -2, 1), (1, 2, 5)$.Free
  4. Q4Find the equation of the line which passes through the point $(1, 2, 3)$ and is parallel to the vector $3\hat{i} + 2\hat{j} - 2\hat{k}$.Preview
  5. Q5Find the equation of the line in vector and in cartesian form that passes through the point with position vector $2\hat{i} - \hat{j} + 4\hat…Preview
  6. Q6Find the cartesian equation of the line which passes through the point $(-2, 4, -5)$ and parallel to the line given by $\frac{x+3}{3} = \fra…Preview
  7. Q7The cartesian equation of a line is $\frac{x-5}{3} = \frac{y+4}{7} = \frac{z-6}{2}$. Write its vector form.Preview
  8. Q8Find the angle between the following pairs of lines: (i) $\vec{r} = 2\hat{i} - 5\hat{j} + \hat{k} + \lambda(3\hat{i} + 2\hat{j} + 6\hat{k})$…Preview
  9. Q9Find the angle between the following pair of lines: (i) $\frac{x-2}{2} = \frac{y-1}{5} = \frac{z+3}{-3}$ and $\frac{x+2}{-1} = \frac{y-4}{8}…Preview
  10. Q10Find the values of $p$ so that the lines $\frac{1-x}{3} = \frac{7y-14}{2p} = \frac{z-3}{2}$ and $\frac{7-7x}{3p} = \frac{y-5}{1} = \frac{6-z…Preview
  11. Q11Show that the lines $\frac{x-5}{7} = \frac{y+2}{-5} = \frac{z}{1}$ and $\frac{x}{1} = \frac{y}{2} = \frac{z}{3}$ are perpendicular to each o…Preview
  12. Q12Find the shortest distance between the lines $\vec{r} = (\hat{i} + 2\hat{j} + \hat{k}) + \lambda(\hat{i} - \hat{j} + \hat{k})$ and $\vec{r}…Preview
  13. Q13Find the shortest distance between the lines $\frac{x+1}{7} = \frac{y+1}{-6} = \frac{z+1}{1}$ and $\frac{x-3}{1} = \frac{y-5}{-2} = \frac{z-…Preview
  14. Q14Find the shortest distance between the lines whose vector equations are $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \lambda(\hat{i} - 3\hat…Preview
  15. Q15Find the shortest distance between the lines whose vector equations are $\vec{r} = (1-t)\hat{i} + (t-2)\hat{j} + (3-2t)\hat{k}$ and $\vec{r}…Preview

Miscellaneous

+Miscellaneous Exercisei5 questions
  1. Q1Find the angle between the lines whose direction ratios are $a, b, c$ and $b-c, c-a, a-b$.Free
  2. Q2Find the equation of a line parallel to x-axis and passing through the origin.Free
  3. Q3If 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, find…Preview
  4. Q4Find the shortest distance between lines $\vec{r} = 6\hat{i} + 2\hat{j} + 2\hat{k} + \lambda (\hat{i} - 2\hat{j} + 2\hat{k})$ and $\vec{r} =…Preview
  5. Q5Find the vector equation of the line passing through the point $(1, 2, -4)$ and perpendicular to the two lines: $\frac{x-8}{3} = \frac{y+19}…Preview

Summary

- Direction Cosines: For a line with angles to axes, direction cosines are , , , with . - Direction Ratios: Any triple proportional to ; if are direction ratios, then , etc.

Exemplar Problems

Higher-order thinking / exemplar-style practice problems.

+Show 24 questions24 questions
  1. Q1Find the position vector of a point $A$ in space such that $\vec{OA}$ is inclined at $60^\circ$ to $OX$ and at $45^\circ$ to $OY$ and $|\vec…Free
  2. Q2Find the vector equation of the line which is parallel to the vector $3\hat{i} - 2\hat{j} + 6\hat{k}$ and which passes through the point $(1…Free
  3. Q3Show that the lines $\dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{4}$ and $\dfrac{x-4}{5} = \dfrac{y-1}{2} = z$ intersect. Also, find their…Free
  4. Q4Find the angle between the lines $\vec{r} = 3\hat{i} - 2\hat{j} + 6\hat{k} + \lambda(2\hat{i} + \hat{j} + 2\hat{k})$ and $\vec{r} = (2\hat{j…Preview
  5. Q5Prove that the line through $A(0, -1, -1)$ and $B(4, 5, 1)$ intersects the line through $C(3, 9, 4)$ and $D(-4, 4, 4)$.Preview
  6. Q6Prove that the lines $x = py + q$, $z = ry + s$ and $x = p'y + q'$, $z = r'y + s'$ are perpendicular if $pp' + rr' + 1 = 0$.Preview
  7. Q7Find the equations of the two lines through the origin which intersect the line $\dfrac{x-3}{2} = \dfrac{y-3}{1} = \dfrac{z}{1}$ at angles o…Preview
  8. Q8Find the angle between the lines whose direction cosines are given by the equations $l + m + n = 0$, $l^2 + m^2 - n^2 = 0$.Preview
  9. Q9If a variable line in two adjacent positions has direction cosines $l, m, n$ and $l + \delta l, m + \delta m, n + \delta n$, show that the s…Preview
  10. Q10Find the foot of perpendicular from the point $(2, 3, -8)$ to the line $\dfrac{4-x}{2} = \dfrac{y}{6} = \dfrac{1-z}{3}$. Also, find the perp…Preview
  11. Q11Find the distance of a point $(2, 4, -1)$ from the line $\dfrac{x+5}{1} = \dfrac{y+3}{4} = \dfrac{z-6}{-9}$.Preview
  12. Q12Find the shortest distance between the lines given by $\vec{r} = (8 + 3\lambda)\hat{i} - (9 + 16\lambda)\hat{j} + (10 + 7\lambda)\hat{k}$ an…Preview
  13. Q13$\vec{AB} = 3\hat{i} - \hat{j} + \hat{k}$ and $\vec{CD} = -3\hat{i} + 2\hat{j} + 4\hat{k}$ are two vectors. The position vectors of the poin…Preview
  14. Q14Show that the straight lines whose direction cosines are given by $2l + 2m - n = 0$ and $mn + nl + lm = 0$ are at right angles.Preview
  15. Q15If $l_1, m_1, n_1$; $l_2, m_2, n_2$; $l_3, m_3, n_3$ are the direction cosines of three mutually perpendicular lines, prove that the line wh…Preview
  16. Q16The direction cosines of the vector $(2\hat{i} + 2\hat{j} - \hat{k})$ are __________.Preview
  17. Q17The vector equation of the line $\dfrac{x-5}{3} = \dfrac{y+4}{7} = \dfrac{z-6}{2}$ is __________.Preview
  18. Q18The vector equation of the line through the points $(3, 4, -7)$ and $(1, -1, 6)$ is __________.Preview
  19. Q19State whether the following statement is True or False: The vector equation of the line $\dfrac{x-5}{3} = \dfrac{y+4}{7} = \dfrac{z-6}{2}$ i…Preview
  20. Q20State whether the following statement is True or False: The equation of a line, which is parallel to $2\hat{i} + 3\hat{j} + \hat{k}$ and whi…Preview
  21. Q21Distance of the point $(\alpha, \beta, \gamma)$ from $y$-axis is (A) $\beta$ (B) $|\beta|$ (C) $|\beta| + |\gamma|$ (D) $\sqrt{\alpha^2 + \g…Preview
  22. Q22If the direction cosines of a line are $k, k, k$, then (A) $k > 0$ (B) $0 < k < 1$ (C) $k = 1$ (D) $k = \dfrac{1}{\sqrt{3}}$ or $-\dfrac{1}{…Preview
  23. Q23The reflection of the point $(\alpha, \beta, \gamma)$ in the $xy$-plane is (A) $(\alpha, \beta, 0)$ (B) $(0, 0, \gamma)$ (C) $(-\alpha, -\be…Preview
  24. Q24The area of the quadrilateral $ABCD$, where $A(0, 4, 1)$, $B(2, 3, -1)$, $C(4, 5, 0)$ and $D(2, 6, 2)$, is equal to (A) $9$ sq. units (B) $1…Preview

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 9 questions9 questions
  1. Q1A bird flies through a distance in a straight line given by the vector 𝑖̂ + 2𝑗̂ + 𝑘̂ . A man standing beside a straight metro rail track giv…Preview
  2. Q2The distance of the point with position vector 3𝑖̂ + 4𝑗̂ + 5𝑘̂ from the y-axis is (A) 4 units (B) √34 units (C) 5 units (D) 5√2 unitsPreview
  3. Q3The value of 𝛼 if the angle between 𝑝⃗ = 2𝛼2𝑖̂ − 3𝛼𝑗̂ + 𝑘̂ and 𝑞⃗ = 𝑖̂ + 𝑗̂ + 𝛼𝑘̂ is obtuse, is (A) 𝑅 − [0, 1] (B) (0, 1) (C) [0, ∞) (D) [1,…Preview
  4. Q4(b) A person standing at $O(0, 0, 0)$ is watching an aeroplane which is at the coordinate point $A(4, 0, 3)$. At the same time he saw a bird…Preview
  5. Q5(a) Find the shortest distance between the lines $l_1$ and $l_2$ whose vector equations are $\vec{r} = (-\hat{i} - \hat{j} - \hat{k}) + \lam…Preview
  6. Q6If the image of the point $P(x, y, z)$ in the line $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$ is $P'(1, 0, 7)$, then find the coordinates…Preview
  7. Q7(a) Find the image $A'$ of the point $A(1, 6, 3)$ in the line $\frac{x}{1} = \frac{y-1}{2} = \frac{z-2}{3}$. Also, find the equation of the…Preview
  8. Q8Find the shortest distance between the lines $\vec{r} = (4\hat{i} - \hat{j}) + \lambda(\hat{i} + 2\hat{j} - 3\hat{k})$ and $\vec{r} = (\hat{…Preview
  9. Q9Find the distance of the point $(-1, -5, -10)$ from the point of intersection of the line $\vec{r} = 2\hat{i} - \hat{j} + 2\hat{k} + \lambda…Preview