Class 12 Maths Three Dimensional Geometry Notes

11.1 Introduction

Three Dimensional Geometry (3D Geometry) deals with points, lines, and planes in space.

• Uses vector and coordinate methods
• Essential in physics, engineering, and computer graphics
• Helps visualize and solve geometrical problems in 3D space

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11.2 Direction Cosines and Direction Ratios of a Line

• Direction cosines: Cosines of angles between a line and the x, y, z axes

$$\cos \alpha, \cos \beta, \cos \gamma$$cosα,cosβ,cosγ

• Must satisfy:

$$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$$cos2α+cos2β+cos2γ=1

• Direction ratios: Any proportional numbers $l, m, n$l,m,n in the direction of the line
• Relationship: Direction cosines are normalized direction ratios

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11.3 Equation of a Line in Space

🔹 1. Vector Form

$$\vec{r} = \vec{a} + \lambda \vec{b}$$r=a+λb

• $\vec{r}$r: Position vector of any point on the line
• $\vec{a}$a: Position vector of a point on the line
• $\vec{b}$b: Direction vector
• $\lambda$λ: Parameter

🔹 2. Cartesian Form

$$\frac{x – x_1}{l} = \frac{y – y_1}{m} = \frac{z – z_1}{n}$$lx−x1​​=my−y1​​=nz−z1​​

• $(x_1, y_1, z_1)$(x1​,y1​,z1​) is a point on the line
• $l, m, n$l,m,n are direction ratios

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11.4 Angle between Two Lines

• If $\vec{b_1} = (l_1, m_1, n_1)$b1​​=(l1​,m1​,n1​) and $\vec{b_2} = (l_2, m_2, n_2)$b2​​=(l2​,m2​,n2​) are direction vectors:

$$\cos \theta = \frac{\vec{b_1} \cdot \vec{b_2}}{|\vec{b_1}| |\vec{b_2}|}$$cosθ=∣b1​​∣∣b2​​∣b1​​⋅b2​​​

• Used to find acute or obtuse angles between lines in space

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11.5 Shortest Distance between Two Lines

• For skew lines (non-parallel, non-intersecting):

$$\text{Distance } d = \frac{|(\vec{r_2} – \vec{r_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|}$$Distance d=∣b1​​×b2​​∣∣(r2​​−r1​​)⋅(b1​​×b2​​)∣​

• $\vec{r_1}, \vec{r_2}$r1​​,r2​​ are points on the two lines
• $\vec{b_1}, \vec{b_2}$b1​​,b2​​ are their direction vectors
• Formula ensures perpendicular distance between the lines

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✅ Key Points to Remember

• 3D geometry studies points, lines, and planes in space
• Direction cosines and ratios describe line orientation
• Equation of a line can be vector form or cartesian form
• Angle between lines uses dot product of direction vectors
• Shortest distance uses cross product for skew lines