In Class 11, the introduction to three-dimensional (3D) geometry builds on the concepts of coordinate geometry learned in earlier classes. Here’s an overview of what you’ll typically study:

1. **Coordinates in Three Dimensions**: In addition to the familiar two-dimensional coordinate system (x-axis and y-axis), a third axis (z-axis) is added to form the three-dimensional coordinate system. A point in 3D space is represented by an ordered triple (x, y, z), where x, y, and z are the distances along the x-axis, y-axis, and z-axis, respectively.

2. **Distance Formula in 3D**: The distance between two points \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\) in 3D space is given by:

\[ \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2} \]

3. **Section Formula in 3D**: The coordinates of a point dividing a line segment joining two points \(P(x_1, y_1, z_1)\) and \(Q(x_2, y_2, z_2)\) in a given ratio \(m:n\) are given by:

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

4. **Direction Cosines and Direction Ratios**: The direction cosines of a line are the cosines of the angles that the line makes with the positive x-axis, y-axis, and z-axis. The direction ratios of a line are the ratios of the differences in the coordinates of two points on the line.

5. **Equation of a Line in 3D**: The equation of a line passing through a point \(P(x_1, y_1, z_1)\) and parallel to a vector \(\vec{v} = \langle a, b, c \rangle\) is given by:

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

6. **Angle between Two Lines**: The angle \(\theta\) between two lines with direction cosines \(l_1, m_1, n_1\) and \(l_2, m_2, n_2\) is given by:

\[ \cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \sqrt{l_2^2 + m_2^2 + n_2^2}} \]

Understanding three-dimensional geometry is important in various fields, including physics, engineering, and computer graphics, where objects and spaces are represented and manipulated in three dimensions.

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