Class 12 Notes - Three Dimensional Geometry

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Three Dimensional Geometry

Three Dimensional Geometry (3D Geometry) deals with the study of points, lines, and planes in space. Unlike two-dimensional geometry, which is limited to a plane, 3D geometry considers the position and relationships of objects in all three spatial dimensions: length (x-axis), width (y-axis), and height (z-axis).

1. Coordinates in Space

  • Any point in space is represented by an ordered triplet (x, y, z).
  • The axes are mutually perpendicular and intersect at the origin (0, 0, 0).
  • The distance of a point from the origin is given by:
    Distance = √(x² + y² + z²)

2. Distance Between Two Points

The distance between points \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\) is:

\( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \)

3. Section Formula

If a point \(P\) divides the line joining \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\) in the ratio \(m:n\), then the coordinates of \(P\) are:

\( P = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n} \right) \)

4. Direction Cosines and Direction Ratios

  • Direction Cosines (l, m, n): Cosines of the angles made by a line with the x, y, and z axes.
  • Direction Ratios: Any set of numbers proportional to the direction cosines.
  • If a line has direction ratios \(a, b, c\), then its direction cosines are:
    \( l = \frac{a}{\sqrt{a^2 + b^2 + c^2}},\quad m = \frac{b}{\sqrt{a^2 + b^2 + c^2}},\quad n = \frac{c}{\sqrt{a^2 + b^2 + c^2}} \)

5. Equation of a Line in Space

  • Vector Form:
    \( \vec{r} = \vec{a} + \lambda \vec{b} \)
    where \( \vec{a} \) is the position vector of a point on the line, \( \vec{b} \) is a direction vector, and \( \lambda \) is a scalar.
  • Cartesian Form:
    \( \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \)

6. Equation of a Plane

  • General Form:
    \( ax + by + cz + d = 0 \)
  • Vector Form:
    \( \vec{r} \cdot \vec{n} = d \)
    where \( \vec{n} \) is a normal vector to the plane.

7. Angle Between Two Lines

If two lines have direction ratios \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\), the angle \( \theta \) between them is:

\( \cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}} \)

8. Angle Between Two Planes

If the normals to the planes are \( (a_1, b_1, c_1) \) and \( (a_2, b_2, c_2) \), then:

\( \cos\theta = \frac{a_1a_2 + b_1b_2 + c_1c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \cdot \sqrt{a_2^2 + b_2^2 + c_2^2}} \)

9. Distance of a Point from a Plane

The distance of point \( (x_1, y_1, z_1) \) from the plane \( ax + by + cz + d = 0 \) is:

\( \frac{|a x_1 + b y_1 + c z_1 + d|}{\sqrt{a^2 + b^2 + c^2}} \)

10. Important Tips

  • Visualize points, lines, and planes in 3D using diagrams.
  • Practice converting between vector and Cartesian forms.
  • Remember the formulas for distances and angles.
  • Draw sketches for better understanding of problems.

Practice Problems

  1. Find the distance between the points (1, 2, 3) and (4, 5, 6).
  2. Find the equation of the plane passing through (1, 2, 3) and perpendicular to the vector \( 2\hat{i} + 3\hat{j} + 4\hat{k} \).
  3. Find the angle between the lines with direction ratios (1, 2, 2) and (2, 1, -2).
  4. Find the coordinates of the point dividing the line joining (2, -1, 3) and (4, 3, 5) in the ratio 2:3.

Summary

  • 3D Geometry extends coordinate geometry to three dimensions.
  • Key concepts: coordinates, distance, section formula, direction cosines/ratios, equations of lines and planes, angles, and distances.
  • Widely used in physics, engineering, and real-world applications.