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Mathematics · Class 12 Science

Ch 9Three-Dimensional Geometry — Class 12 Mathematics, concept-first.

Coordinate geometry, as studied in two dimensions, describes every point of a plane by an ordered pair measured against two mutually perpendicular reference lines, the -axis and the -axis, meeting at a fixed origin .

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Key concepts

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Direction Cosines and Direction Ratios

The direction angles alpha, beta, gamma of a non-zero vector are the angles it makes with the positive X, Y and Z axes; their cosines l=cos(alpha), m=cos(beta), n=cos(gamma) are called its direction cosines, and they alw…

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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.

1

Introduction to 3D Coordinate Geometry

Coordinate geometry, as studied in two dimensions, describes every point of a plane by an ordered pair measured against two mutually perpendicular reference lines, the -axis and the -axis, meeting at…

2

Coordinate Axes and Coordinate Planes

To locate a point in space, three mutually perpendicular straight lines are drawn through a fixed point , called the origin.

3

Coordinates of a Point and Distance Between Two Points

Coordinates of a point. Once the three coordinate axes and origin are fixed (Section 2), every point of space corresponds to a unique ordered triple of real numbers -- its coordinates -- and conversel…

4

Direction Cosines and Direction Ratios

A line in space has a definite length only between two specified points, but it has a direction regardless of length -- and describing that direction precisely, independent of where the line is positi…

5

Equation of a Line in Space (Cartesian and Vector Form)

A straight line in space is completely determined by (a) one point on it and its direction, or (b) any two distinct points on it (which together fix both a point and, via their difference, a direction…

6

Skew Lines

Definition. Two straight lines in space are called skew lines if they satisfy both of the following conditions simultaneously: (i) they are not parallel (their direction vectors are not scalar multipl…

7

Shortest Distance Between Two Lines

What "shortest distance" means. Given two lines and in space, the shortest distance between them is the length of the shortest possible line segment joining a point of to a point of -- equivalently, t…

8

Angle Between Two Lines

Definition. The angle between two lines in space, with direction vectors and , is defined as the acute (or right) angle between their directions -- taking whichever of the two supplementary angles for…

Summary

Coordinate axes and planes. Space is referenced against three mutually perpendicular axes meeting at the origin (a right-handed system), giving three coordinate planes (, , ) and eight octants; a poin…

Sample & Board Papers

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

+Show 25 questions25 questions
  1. Q1Find the equation of the plane which passes through (-2, 1, 3) and also through the intersection of the planes 2x - 7y + 4z = 0 and 3x - 5y…Preview
  2. Q2Find the point on the line (x+2)/3 = (y+1)/2 = (z-3)/2 at a distance 3 sqrt(2) units from the point (1, 2, 3).Preview
  3. Q3The angle between the two planes x - y + 2z = 9 and 2x + y + z = 7 is (a) 30 degrees (b) 45 degrees (c) 60 degrees (d) 90 degreesPreview
  4. Q4Show that the line joining (1, -1, 2), (3, 4, -2) is perpendicular to the line through (0, 3, 2) and (3, 5, 6).Preview
  5. Q5Find the equation of the plane passing through (1, 0, 0), (0, 2, 0) and (0, 0, 4).Preview
  6. Q6Find the equation of the line that passes through the origin and (5, 2, 4).Preview
  7. Q7Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to t…Preview
  8. Q8Find the cartesian equation of the line which passes through (1, 2, 3) and parallel to the line (x+3)/3 = (y-4)/5 = (z+8)/6.Preview
  9. Q9Find the coordinates of the foot of the perpendicular from origin to the plane x + y + z = 3.Preview
  10. Q10A plane has the intercepts on axes are a, b, c respectively and p be the perpendicular distance from origin to the plane. Show that 1/a² + 1…Preview
  11. Q11If A = (1, 0, 2) and B = (0, 1, 1), then direction cosines of the line AB are: **OR** Vector a = î + 3ĵ - k̂ and vector b = 2î + 6ĵ + λk̂. I…Preview
  12. Q12The equation of the plane with intercepts 2, 3, 4 units on the x-axis, y-axis and z-axis respectively is: (a) 6x + 4y + 3z = 12 (b) 4x + 3y…Preview
  13. Q13The angle between the straight lines (x-4)/2 = (y-5)/0 = (z-6)/0 and (3-x)/3 = (y-7)/0 = (z-3)/0 is (a) -π (b) -π/2 (c) π (d) π/3Preview
  14. Q14Find direction cosine of the line passing through the points (2, 3, -4) and (1, -2, 3).Preview
  15. Q15Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x - 3y + 4z = 6.Preview
  16. Q16Find the coordinates of the point where the line through the points A = (3,4,1) and B = (5,1,6) crosses the XY plane.Preview
  17. Q17Find the acute angle between the lines whose direction ratios are 2, 1, -2 and 3, -4, 5.Preview
  18. Q18Find the equation of a straight line which passes through the point (1, 2, 3) and is perpendicular to both of the lines x/2 = y/1 = z/3 and…Preview
  19. Q19Find the equation of the plane which passes through the point (-1, -1, 2) and is perpendicular to the planes 3x + 2y - 3z = 1 and 5x - 4y +…Preview
  20. Q20The coordinates of the point where the straight line (x+3)/-1 = (y-2)/3 = (z+2)/2 intersects the xy plane are (a) (0,-5,0) (b) (-4,5,0) (c)…Preview
  21. Q21Given the straight lines (x-2)/a = (y+3)/6 = (z-2)/5 and (x+2)/3 = (y-1)/2a = (z+3)/5, then determine the values of 'a' for which the lines…Preview
  22. Q22Find 'x' such that the four points A(3,2,1), B(4,x,5), C(4,2,-2) and D(6,5,-1) lie on the same plane.Preview
  23. Q23Find the image of the point (1,6,3) with respect to the line x/1 = (y-1)/2 = (z-2)/3, and also find the equation of the line passing through…Preview
  24. Q24If the two straight lines $\dfrac{x-2}{3} = \dfrac{y+1}{-2\lambda} = \dfrac{z-2}{0}$ and $\dfrac{x-1}{1} = \dfrac{2y+3}{3\lambda} = \dfrac{z…Preview
  25. Q25The direction cosines of two straight lines satisfy the conditions $a^2 l + b^2 m + c^2 n = 0$ and $mn + nl + lm = 0$. Show that the two str…Preview

More questions

34 Q
+Show 3 questions3 questions
  1. Q32A 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…Free
  2. Q33Find the direction cosines of the line that is perpendicular to both the lines with direction ratios $1,-2,-2$ and $0,2,1$. Hence find the C…Preview
  3. Q34In computing the shortest distance between two skew lines using $d=\left|\dfrac{(\vec a_2-\vec a_1)\cdot(\vec b_1\times\vec b_2)}{|\vec b_1\…Preview
+Show 9 questions9 questions
  1. Example 1Find the distance between the points $A(1,-2,3)$ and $B(4,2,-1)$.Free
  2. Example 2Show that the points $A(1,2,3)$, $B(3,4,5)$ and $C(-1,0,1)$ are collinear.Free
  3. Example 3Find the direction cosines of the line joining the points $A(1,1,1)$ and $B(3,4,7)$.Free
  4. Example 4The direction ratios of a line are $2,-3,6$. Find its direction cosines.Preview
  5. Example 5Find the vector equation of the line passing through the point $(1,2,3)$ and parallel to the vector $2\hat i-\hat j+2\hat k$. Also write its…Preview
  6. Example 6Find the vector and Cartesian equations of the line passing through the points $A(2,-1,3)$ and $B(5,2,7)$.Preview
  7. Example 7Determine whether the lines through the point $(1,2,3)$ with direction ratios $2,3,4$ and through the point $(4,1,0)$ with direction ratios…Preview
  8. Example 8Show that the line $\vec r=(\hat i+2\hat j+3\hat k)+\lambda(\hat i+\hat j+\hat k)$ and the line $\vec r=(-\hat j+2\hat k)+\mu(2\hat i+\hat j…Preview
  9. Example 9Find the angle between the lines $\vec r=\lambda(2\hat i+2\hat j+\hat k)$ and $\vec r=\mu(4\hat i+\hat j+8\hat k)$.Preview
+Show 4 questions4 questions
  1. Q10Find the distance between the points $P(3,-2,5)$ and $Q(-1,2,1)$.Free
  2. Q11Find the distance of the point $A(2,3,-4)$ from the origin.Free
  3. Q12Show that the points $A(1,2,3)$, $B(-1,-1,-1)$ and $C(3,5,7)$ are collinear.Preview
  4. Q13If the distance between the points $A(k,2,1)$ and $B(2,-1,3)$ is $\sqrt{14}$, find the value(s) of $k$.Preview
+Show 4 questions4 questions
  1. Q14Find the direction cosines of the line joining the points $A(1,1,1)$ and $B(7,3,4)$.Free
  2. Q15The direction ratios of a line are $4,-4,7$. Find its direction cosines.Free
  3. Q16A line makes angles of $90^\circ$, $60^\circ$ and $30^\circ$ with the $x$-, $y$- and $z$-axes respectively. Find its direction cosines.Preview
  4. Q17A line makes equal angles with the three coordinate axes. Find its direction cosines.Preview
+Show 5 questions5 questions
  1. Q18Find the vector and Cartesian equations of the line passing through the point $(2,-3,4)$ and parallel to the vector $\hat i+2\hat j-2\hat k$…Free
  2. Q19Find the Cartesian equation of the line passing through the points $(1,-2,3)$ and $(4,0,-3)$.Free
  3. Q20Find the vector equation of the line whose Cartesian equation is $\dfrac{x-3}{2}=\dfrac{y+2}{-3}=\dfrac{z-5}{6}$.Preview
  4. Q21Find the vector equation of the line passing through the points $A(1,0,2)$ and $B(3,4,-1)$.Preview
  5. Q22Find the coordinates of the point where the line $\dfrac{x-1}{2}=\dfrac{y+1}{3}=\dfrac{z-2}{4}$ crosses the $XY$-plane.Preview
+Show 5 questions5 questions
  1. Q23Show that the lines $\vec r=(\hat i+\hat j)+\lambda(2\hat i-\hat j+\hat k)$ and $\vec r=(2\hat i+\hat j-\hat k)+\mu(3\hat i-5\hat j+2\hat k)…Free
  2. Q24Find the shortest distance between the lines $\vec r=\lambda(\hat i+\hat j)$ and $\vec r=4\hat k+\mu(\hat i-\hat j)$.Free
  3. Q25Show that the lines $\dfrac{x-2}{1}=\dfrac{y-3}{2}=\dfrac{z-1}{-1}$ and $\dfrac{x-2}{3}=\dfrac{y-3}{-1}=\dfrac{z-1}{2}$ intersect, and find…Preview
  4. Q26Show that the line through the point $(1,0,-1)$ with direction ratios $2,1,1$ and the line through the point $(0,2,3)$ with direction ratios…Preview
  5. Q27Find the shortest distance between the parallel lines $\vec r=(\hat i+2\hat j+3\hat k)+\lambda(2\hat i+3\hat j+4\hat k)$ and $\vec r=(2\hat…Preview
+Show 4 questions4 questions
  1. Q28Find the angle between the lines whose direction ratios are $1,1,0$ and $1,0,1$.Free
  2. Q29Find the angle between the lines $\vec r=(\hat i+2\hat j)+\lambda(2\hat i+\hat j+2\hat k)$ and $\vec r=3\hat k+\mu(3\hat i+2\hat j+6\hat k)$…Free
  3. Q30Find the value of $\lambda$ so that the lines with direction ratios $3,2\lambda,2$ and $3\lambda,1,-5$ are perpendicular to each other.Preview
  4. Q31Show that the lines whose direction ratios are $1,2,3$ and $2,-4,2$ are perpendicular to each other.Preview