Three Dimensional Geometry

Three Dimensional Geometry

Class 11 me 3D me points dekhe the. Ab vectors ki madad se 3D me lines aur planes ke equations, aur unke beech ke angles aur distances padhenge.

1. Direction Cosines aur Direction Ratios

• Direction cosines (DCs): line jo axes ke saath \(\alpha, \beta, \gamma\) angle banaye, uske \(l = \cos\alpha\), \(m = \cos\beta\), \(n = \cos\gamma\). Property: \(l^{2} + m^{2} + n^{2} = 1\).
• Direction ratios (DRs): DCs ke proportional koi bhi \(a, b, c\). DCs nikaalne ke liye normalize karo: \(l = \dfrac{a}{\sqrt{a^{2}+b^{2}+c^{2}}}\), etc.

2. Equation of a Line

Point \(\vec{a}\) se guzarti aur direction \(\vec{b}\) wali line:

Vector form:

\[ \vec{r} = \vec{a} + \lambda\,\vec{b}. \]

Cartesian form (point \((x_1, y_1, z_1)\), DRs \(a, b, c\)):

\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}. \]

Angle between two lines

Direction vectors \(\vec{b_1}, \vec{b_2}\) ke beech:

\[ \cos\theta = \frac{|\vec{b_1}\cdot\vec{b_2}|}{|\vec{b_1}|\,|\vec{b_2}|}. \]

Perpendicular: \(\vec{b_1}\cdot\vec{b_2} = 0\); Parallel: \(\vec{b_1} \parallel \vec{b_2}\).

3. Equation of a Plane

Vector form (normal \(\vec{n}\), distance \(d\)):

\[ \vec{r}\cdot\hat{n} = d. \]

Cartesian form:

\[ ax + by + cz + d = 0, \]

jahan \((a, b, c)\) plane ke normal ke DRs hain.

Angle between two planes

Normals \(\vec{n_1}, \vec{n_2}\) ke beech ka angle hi planes ke beech ka angle:

\[ \cos\theta = \frac{|\vec{n_1}\cdot\vec{n_2}|}{|\vec{n_1}|\,|\vec{n_2}|}. \]

4. Distance of a Point from a Plane

Point \((x_1, y_1, z_1)\) ki plane \(ax + by + cz + d = 0\) se doori:

\[ D = \frac{|a x_1 + b y_1 + c z_1 + d|}{\sqrt{a^{2} + b^{2} + c^{2}}}. \]

Key Takeaways

• DCs: \(l^{2}+m^{2}+n^{2}=1\); DRs unke proportional.
• Line: \(\vec{r} = \vec{a} + \lambda\vec{b}\) ya \(\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\).
• Lines/planes ke beech angle dot product se (\(\cos\theta\)).
• Point-to-plane distance: \(\frac{|ax_1+by_1+cz_1+d|}{\sqrt{a^{2}+b^{2}+c^{2}}}\).