📌 1. Introduction

In 3D geometry, every point is represented by three coordinates:$$(x, y, z)$$(x,y,z)

• $x$x → distance along x-axis
• $y$y → distance along y-axis
• $z$z → distance along z-axis

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📖 2. Coordinate Axes and Planes

• X-axis, Y-axis, Z-axis
• XY-plane, YZ-plane, ZX-plane

These divide space into 8 octants.

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📐 3. Distance Between Two Points

If points are:$$P(x_1, y_1, z_1), \quad Q(x_2, y_2, z_2)$$P(x1​,y1​,z1​),Q(x2​,y2​,z2​)

Distance:

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$PQ = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$PQ=(x2​−x1​)2+(y2​−y1​)2+(z2​−z1​)2​

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📌 4. Section Formula

If a point divides the line joining two points in ratio $m:n$m:n:

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$$\vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n}$$r=m+nmb+na​

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$\vec{r} = \frac{m\vec{b} + n\vec{a}}{m+n}$r=m+nmb+na​

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🔹 Coordinates Form:

$$x = \frac{mx_2 + nx_1}{m+n}, \quad y = \frac{my_2 + ny_1}{m+n}, \quad z = \frac{mz_2 + nz_1}{m+n}$$x=m+nmx2​+nx1​​,y=m+nmy2​+ny1​​,z=m+nmz2​+nz1​​

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📌 5. Direction Cosines and Ratios

🔹 Direction Cosines (l, m, n)

$$l^2 + m^2 + n^2 = 1$$l2+m2+n2=1

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$l^2 + m^2 + n^2 = 1$l2+m2+n2=1

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🔹 Direction Ratios

Any proportional values to direction cosines.

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📐 6. Equation of a Line in Space

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🔹 Vector Form:

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

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

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🔹 Cartesian Form:

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

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

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📌 7. Angle Between Two Lines

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$$\cos \theta = \frac{l_1l_2 + m_1m_2 + n_1n_2}{\sqrt{l_1^2+m_1^2+n_1^2} \cdot \sqrt{l_2^2+m_2^2+n_2^2}}$$cosθ=l12​+m12​+n12​​⋅l22​+m22​+n22​​l1​l2​+m1​m2​+n1​n2​​

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$\cos \theta = \frac{l_1l_2 + m_1m_2 + n_1n_2}{\sqrt{l_1^2+m_1^2+n_1^2} \cdot \sqrt{l_2^2+m_2^2+n_2^2}}$cosθ=l12​+m12​+n12​​⋅l22​+m22​+n22​​l1​l2​+m1​m2​+n1​n2​​

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📌 Special Cases

• If $\theta = 0^\circ$θ=0∘ → parallel
• If $\theta = 90^\circ$θ=90∘ → perpendicular

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📐 8. Shortest Distance Between Two Lines

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$$d = \frac{|(\vec{a_2} – \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|}$$d=∣b1​​×b2​​∣∣(a2​​−a1​​)⋅(b1​​×b2​​)∣​

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$d = \frac{|(\vec{a_2} – \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|}$d=∣b1​​×b2​​∣∣(a2​​−a1​​)⋅(b1​​×b2​​)∣​

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📌 9. Equation of a Plane

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🔹 General Form:

$$ax + by + cz + d = 0$$ax+by+cz+d=0

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$ax + by + cz + d = 0$ax+by+cz+d=0

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🔹 Vector Form:

$$\vec{r} \cdot \vec{n} = d$$r⋅n=d

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$\vec{r} \cdot \vec{n} = d$r⋅n=d

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📐 10. Angle Between Two Planes

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$$\cos \theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \cdot \sqrt{a_2^2+b_2^2+c_2^2}}$$cosθ=a12​+b12​+c12​​⋅a22​+b22​+c22​​a1​a2​+b1​b2​+c1​c2​​

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$\cos \theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \cdot \sqrt{a_2^2+b_2^2+c_2^2}}$cosθ=a12​+b12​+c12​​⋅a22​+b22​+c22​​a1​a2​+b1​b2​+c1​c2​​

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📌 11. Distance of Point from Plane

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$$d = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$$d=a2+b2+c2​∣ax1​+by1​+cz1​+d∣​

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$d = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}}$d=a2+b2+c2​∣ax1​+by1​+cz1​+d∣​

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📊 12. Applications of 3D Geometry

• Navigation systems
• Computer graphics
• Robotics
• Architecture
• Physics

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❗ Common Mistakes

• Sign errors in formulas
• Confusing direction ratios
• Mistakes in distance formula
• Wrong substitution

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🧠 Exam Tips

• Learn all formulas
• Practice diagrams
• Solve NCERT problems
• Focus on vectors

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📚 Practice Questions

1. Find distance between two points
2. Equation of line
3. Angle between lines/planes
4. Shortest distance problems

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🎯 Conclusion

Three Dimensional Geometry helps us understand spatial relationships and solve real-world problems involving space. Mastering formulas and concepts is essential for scoring high in exams.