Three Dimensional Geometry Class 12th Notes - Free NCERT Class 12 Maths Chapter 11 Notes

Three Dimensional Geometry Class 12th Notes - Free NCERT Class 12 Maths Chapter 11 Notes - Download PDF

Edited By Komal Miglani | Updated on Apr 04, 2025 10:20 PM IST

Notes for Class 12 Maths chapter 11 are regarding Three-Dimensional Geometry. This Class 12 maths chapter 11 notes contains direction cosines, direction ratios, equation of a straight line in different forms, the angle between the lines, Equation of parallel and perpendicular lines and their distances, intercept form. Three-dimensional geometry deals with the study of points, lines, planes, and shapes in space. Unlike two-dimensional geometry, it involves three axes: x, y, and z, which represent length, breadth, and height. It helps us understand the position and distance between objects, angles between lines and planes, and equations of 3D objects. In real life, 3D geometry is used in architecture, aviation, robotics, and 3D modeling.

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NCERT Class 12 Chapter 11 Notes
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Important points to note:

NCERT class 12th Math chapter 11 notes contain standard formulas and detailed information that are used to solve the questions. NCERT Class 12 Math Chapter 11 notes and Class 12 Math Chapter 11 notes contain a detailed explanation of topics, examples, and exercises. The document will help students cover all the topics that are in the NCERT Notes for Class 12 Math chapter 11 textbook. All these concepts can be downloaded as PDF from Class 12 Maths chapter 11 notes pdf download, NCERT notes for Class 12 Maths chapter 11 are the notes for Three-dimensional geometry, Class 12 Three-dimensional geometry notes pdf download.

NCERT Class 12 Chapter 11 Notes

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A line OA makes angles α,β,γ with the x,y, and z-axis, respectively. Then cosα, cosβ, and cosγ are called direction cosines.

l = cosα, m = cosβ, n = cosγ

$\begin{aligned} l^{2} + m^{2} + n^{2} = 1 \\ \cos^{2} α + \cos^{2} β + \cos^{2} γ = 1 \end{aligned}$

Note: Direction cosines are always unique.

Direction ratios: The values that are proportional to directional cosines are called direction ratios.

$a, b, c$ are ratios, then $\frac{l}{a} = \frac{m}{b} = \frac{n}{c}$
Direction cosines of ratios a,b,c :

$\begin{aligned} l = \pm \frac{a}{\sqrt{a^{2} + b^{2} + c^{2}}} \\ m = \pm \frac{b}{\sqrt{a^{2} + b^{2} + c^{2}}} \\ n = \pm \frac{c}{\sqrt{a^{2} + b^{2} + c^{2}}} \end{aligned}$

Direction cosines pass through two points:

$P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$

$\cos α = \frac{x_{2} - x_{1}}{O A}, \cos β = \frac{y_{2} - y_{1}}{O A}, \cos γ = \frac{z_{2} - z_{1}}{O A}$

Direction cosines: $\frac{x_{2} - x_{1}}{O A}, \frac{y_{2} - y_{1}}{O A}, \frac{z_{2} - z_{1}}{O A}$

Direction ratios: $x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1}$

NOTE: Directions ratios need not be unique

Equation of Straight Line:

Equation of a Line passing through the Point and parallel to vector b

$\vec{r} = \vec{a} + λ \vec{b}$

where, $\vec{a}$ is a position vector. $\vec{b}$ is a vector that is parallel to the line

Cartesian form:

$\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c}$

Where, $(x_{1}, y_{1}, z_{1})$ be the point that line passes through and $a, b, c$ are the direction ratios.

l,m,n are the direction cosines, then the equation is:

$\frac{x - x_{1}}{l} = \frac{y - y_{1}}{m} = \frac{z - z_{1}}{n}$

Equation of line that passes through 2 points:

Points are : $(x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2})$

$\vec{r} = \vec{a} + λ (\vec{b} - \vec{a})$

$\vec{a}, \vec{b}$ are a position vectors.

Cartesian form of two points:

Points are : $(x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2})$

$\frac{x - x_{1}}{x_{1} - x_{2}} = \frac{y - y_{1}}{y_{1} - y_{2}} = \frac{z - z_{1}}{z_{1} - z_{2}}$

Vector equations of 2 lines:

$\vec{r_{1}} = \vec{a_{1}} + λ (\vec{b_{1}} - \vec{a_{1}})$ and $\vec{r_{2}} = \vec{a_{2}} + λ (\vec{b_{2}} - \vec{a_{2}})$

The angle between 2 lines:

$\cos θ = \frac{b_{1} \cdot b_{2}}{| b_{1} | \cdot | b_{2} |}$

Cartesian form: Let be an angle between the lines below:

$\frac{x - x_{1}}{a_{1}} = \frac{y - y_{1}}{b_{1}} = \frac{z - z_{1}}{c_{1}}$ and $\frac{x - x_{2}}{a_{2}} = \frac{y - y_{2}}{b_{2}} = \frac{z - z_{2}}{c_{2}}$

Then,

$\begin{aligned} \cos θ = | \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}} | \\ \sin θ = \sqrt{1 - \cos^{2} θ} \\ \sin θ = \frac{\sqrt{{(a_{1} b_{2} - a_{2} b_{1})}^{2} + {(b_{1} c_{2} - b_{2} c_{1})}^{2} + {(c_{1} a_{2} - c_{2} a_{1})}^{2}}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}} \end{aligned}$

Direction cosines with angles are:

$\begin{aligned} \cos θ = l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} \\ \sin θ = \sqrt{{(m_{1} n_{2} - m_{2} n_{1})}^{2} + {(n_{1} l_{2} - n_{2} l_{1})}^{2} + {(l_{1} m_{2} - l_{2} m_{1})}^{2}} \end{aligned}$

Few conditions:

When lines are perpendicular, $θ = 90^{\circ}$

Then Cartesian form:

$\begin{aligned} a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0 \\ l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0 \end{aligned}$

When lines are parallel $θ = 0^{\circ}$ .

Then Cartesian form:

$\begin{aligned} \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}} \\ \frac{l_{1}}{l_{2}} = \frac{m_{1}}{m_{2}} = \frac{n_{1}}{n_{2}} \end{aligned}$

Shortest path:

Vector equations of 2 lines:

$\vec{r_{1}} = \vec{a_{1}} + λ \vec{b_{1}}$ and $\vec{r_{2}} = \vec{a_{2}} + μ \vec{b_{2}}$

Shortest distance

$d = | \frac{(b_{1} \times b_{2}) \cdot (a_{2} \times a_{1})}{(b_{1} \times b_{2})} |$

where $\vec{a_{1}}, \vec{a_{2}}$ are position vectors and $\vec{b_{1}}, \vec{b_{2}}$ are vectors in the direction of a line.

Cartesian form for two lines:

$\frac{x - x_{1}}{a_{1}} = \frac{y - y_{1}}{b_{1}} = \frac{z - z_{1}}{c_{1}}$ and $\frac{x - x_{2}}{a_{2}} = \frac{y - y_{2}}{b_{2}} = \frac{z - z_{2}}{c_{2}}$

Cartesian form :

Shortest distance between the lines:

$\begin{aligned} l_{1} : \frac{x - x_{1}}{a_{1}} = \frac{y - y_{1}}{b_{1}} = \frac{z - z_{1}}{c_{1}} \\ l_{2} : \frac{x - x_{2}}{a_{2}} = \frac{y - y_{2}}{b_{2}} = \frac{z - z_{2}}{c_{2}} \end{aligned}$

$| \frac{| \begin{array}{ccc} x_{2} - x_{1} & y_{2} - y_{1} & z_{2} - z_{1} \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array} |}{\sqrt{{(b_{1} c_{2} - b_{2} c_{1})}^{2} + {(c_{1} a_{2} - c_{2} a_{1})}^{2} + {(a_{1} b_{2} - a_{2} b_{1})}^{2}} ∣} |$

Distance between two Parallel Lines:

Lines are said to be coplanar when they are parallel.

Vector equations of 2 lines:

$\vec{r_{1}} = \vec{a_{1}} + λ \vec{b_{1}}$ and $\vec{r_{2}} = \vec{a_{2}} + μ \vec{b_{2}}$

Distance between two Parallel Lines:

$| \frac{(b_{1}) \times (a_{2} - a_{1})}{| b_{2} |} |$

Note: If lines are parallel, they always have the same direction ratios.

Distance between two points: $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$

$| P Q | = \sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2} + {(z_{2} - z_{1})}^{2}}$

The midpoint of two points: $P (x_{1}, y_{1}, z_{1})$ and $Q (x_{2}, y_{2}, z_{2})$

$P Q = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}, \frac{z_{1} + z_{2}}{2})$

PLANE:

The plane is found unique, only if it satisfies any one of the following:

If normal form and distance from the origin are given
pass-through point and perpendicular
passes through non-collinear points

Equation of a plane in normal form:

$\vec{r} \cdot \vec{n} = d$

Cartesian form: one equation of the plane is ax + by + cz = d, and another equation of the plane is lx + my + nz = p

Formula: Foot of perpendicular (ld, md, nd).

Equation of a plane perpendicular to a vector and passing through a point:

Vector equation: $(\vec{r} - \vec{a}) \cdot \vec{n} = 0$

Cartesian form: Equation of a plane that passes through the point $(x_{1}, y_{1}, z_{1})$

$a (x - x_{1}) + b (y - y_{1}) + c (z - z_{1}) = 0$

Equation of a plane passing through non-collinear points:

Vector form: $(\vec{r - a}) \cdot {(\vec{b - a}) \times (\vec{c -} \vec{a})} = 0$

Cartesian form: $(x_{1}, y_{1}, z_{1}), (x_{2}, y_{2}, z_{2})$ and $(x_{3}, y_{3}, z_{3})$ are non-collinear points

Equation: $| \begin{array}{ccc} x - x_{1} & y - y_{1} & z - z_{1} \\ x_{2} - x_{1} & y_{2} - y_{1} & z_{2} - z_{1} \\ x_{3} - x_{1} & y_{3} - y_{1} & z_{3} - z_{1} \end{array} | = 0$

If they are collinear: $| \begin{array}{lll} x_{1} & y_{1} & z_{1} \\ x_{2} & y_{2} & z_{2} \\ x_{3} & y_{3} & z_{3} \end{array} | = 0$

Intercept Form:

Let a, b, and c be the x-intercept, y-intercept, and z-intercept, respectively, then

Equation of intercept: xa + by + zc = 1

Equation of Plane Passing through the intersection of two Planes: The equation of the planes is

$\vec{r_{1}} \cdot \vec{n_{1}} = d_{1}$ and $\vec{r_{2}} \cdot \vec{n_{2}} = d_{2}$

Equation of the plane passing through the intersection :

$r (n_{1} + λ n_{2}) = d_{1} + λ d_{2}$

Cartesian form: equation of planes are $a_{1} x + b_{1} y + c_{1} z = d_{1}$ and $a_{2} x + b_{2} y + c_{2} z = d_{2}$

Equation of the plane passing through the intersection :

$a_{1} x + b_{1} y + c_{1} z - d_{1} + λ (a_{2} x + b_{2} y + c_{2} z - d_{2}) = 0$

Coplanarity:

Lines $: \vec{r_{1}} = \vec{a_{1}} + λ \vec{b_{1}}$ and $\vec{r_{2}} = \vec{a_{2}} + μ \vec{b_{2}}$ are coplanar then Vectorform : $(\vec{a_{1}} - \vec{a_{2}}) \cdot (\vec{b_{1}} \times \vec{b_{2}}) = 0$

Cartesian form:

For Two lines

$\frac{x - x_{1}}{a_{1}} = \frac{y - y_{1}}{b_{1}} = \frac{z - z_{1}}{c_{1}}$ and $\frac{x - x_{2}}{a_{2}} = \frac{y - y_{2}}{b_{2}} = \frac{z - z_{2}}{c_{2}}$ are coplanar then

$| \begin{array}{ccc} x_{2} - x_{1} & y_{2} - y_{1} & z_{2} - z_{1} \\ a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \end{array} | = 0$

The angle between Two Planes: θ is the angle between two planes.

$n_{1}, n_{2}$ are normals, the angle between $\vec{r_{1}} \cdot \vec{n_{1}} = d_{1}$ and $\vec{r_{2}} \cdot \vec{n_{2}} = d_{2}$

$\cos θ = | \frac{n_{1} \cdot n_{2}}{| n_{1} | \cdot | n_{2} |} |$

Cartesian form: planes are $a_{1} x + b_{1} y + c_{1} z = d_{1}$ and $a_{2} x + b_{2} y + c_{2} z = d_{2}$ ,

$\cos θ = \frac{a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}} \sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}}$

Angle Between Line and Plane:

Vector form: The equation of a line is $\vec{r} = (\vec{a} + λ \vec{b})$

angle θ between the line and the normal to the plane is

$\cos θ = \frac{\vec{b \cdot n}}{| \vec{b} \cdot | \vec{n} ∣}$

Cartesian form: a, b, and c are direction ratios, and lx + my + nz + d = 0 is the equation of the plane.

$\sin θ = \frac{a l + b m + c n}{\sqrt{a^{2} + b^{2} + c^{2}} \sqrt{l^{2} + m^{2} + n^{2}}}$

NCERT Class 12 Notes Chapter Wise

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NCERT Class 12 Maths Chapter 11 Notes

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Important points to note:

NCERT problems are very important in order to perform well in exams. Students must try to solve all the NCERT problems, including miscellaneous exercises, and if needed, refer to the NCERT Solutions for Class 12 Maths Chapter 11 Three-Dimensional Geometry.
Students are advised to go through the NCERT Class 12 Maths Chapter 11 Notes before solving the questions.
To boost your exam preparation as well as for quick revision, these NCERT notes are very useful.

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Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

A person trying to lose weight by burning fat lifts a mass of 10 kg upto a height of 1 m 1000 times. Assume that the potential energy lost each time he lowers the mass is dissipated. How much fat will he use up considering the work done only when the weight is lifted up ? Fat supplies 3.8×10⁷J of energy per kg which is converted to mechanical energy with a 20% efficiency rate. Take g = 9.8 ms⁻² :

Option 1)

2.45×10⁻³ kg

Option 2)

6.45×10⁻³ kg

Option 3)

9.89×10⁻³ kg

Option 4)

12.89×10⁻³ kg

An athlete in the olympic games covers a distance of 100 m in 10 s. His kinetic energy can be estimated to be in the range

Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

A particle is projected at 60⁰ to the horizontal with a kinetic energy $K$ . The kinetic energy at the highest point

Option 1) $K/2\,$	Option 2) $\; K\;$
Option 3) $zero\;$	Option 4) $K/4$

In the reaction,

$2Al_{(s)}+6HCL_{(aq)}\rightarrow 2Al^{3+}\, _{(aq)}+6Cl^{-}\, _{(aq)}+3H_{2(g)}$

Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

How many moles of magnesium phosphate, $Mg_{3}(PO_{4})_{2}$ will contain 0.25 mole of oxygen atoms?

Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

If we consider that 1/6, in place of 1/12, mass of carbon atom is taken to be the relative atomic mass unit, the mass of one mole of a substance will

Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

With increase of temperature, which of these changes?

Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

Number of atoms in 558.5 gram Fe (at. wt.of Fe = 55.85 g mol^-1) is

Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
Option 3) half that in 8 g He	Option 4) 558.5 × 6.023 × 10²³

A pulley of radius 2 m is rotated about its axis by a force F = (20t - 5t²) newton (where t is measured in seconds) applied tangentially. If the moment of inertia of the pulley about its axis of rotation is 10 kg m² , the number of rotations made by the pulley before its direction of motion if reversed, is