12th Mathematics (Arts & Science) Part 1 Chapter 6 Solution (Digest) Maharashtra state board

Chapter 6 Line and Plane

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Line in Mathematics

A line in mathematics is a fundamental geometric concept representing a straight one-dimensional figure extending infinitely in both directions. It has no thickness and is defined by at least two points.

Key Characteristics and Definitions:

1. Representation:

   • Two-Point Form: A line can be uniquely determined by any two distinct points on it. If the points are $A(x_1, y_1)$ and $B(x_2, y_2)$, the line can be expressed in the form: $$\frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1}$$
   • Slope-Intercept Form: A line can be expressed as $y = mx + c$, where $m$ is the slope (indicating the steepness) and $c$ is the y-intercept (the point where the line crosses the y-axis).
   • Point-Slope Form: Given a point $(x_1, y_1)$ and a slope $m$, the line can be written as $y - y_1 = m(x - x_1)$.
   • General Form: $Ax + By + C = 0$, where $A$, $B$, and $C$ are constants.

2. Slope:

   • The slope of a line is given by $m = \frac{y_2 - y_1}{x_2 - x_1}$. It measures the rate at which $y$ changes with respect to $x$.

3. Properties:

   • Lines are straight and extend infinitely in both directions.
   • Parallel lines have the same slope but different y-intercepts.
   • Perpendicular lines have slopes that are negative reciprocals of each other (i.e., $m_1 \cdot m_2 = -1$).

Plane in Mathematics

A plane in mathematics is a two-dimensional flat surface that extends infinitely in all directions. It is defined by at least three non-collinear points (points not lying on the same line).

Key Characteristics and Definitions:

1. Representation:

   • Point-Normal Form: A plane can be expressed using a point $(x_1, y_1, z_1)$ on the plane and a normal vector $\vec{n} = (A, B, C)$. The equation of the plane is: $$A(x - x_1) + B(y - y_1) + C(z - z_1) = 0$$
   • General Form: The equation of a plane can be written as $Ax + By + Cz + D = 0$, where $A$, $B$, $C$, and $D$ are constants.

2. Normal Vector:

   • The normal vector $\vec{n} = (A, B, C)$ is perpendicular to the plane and is crucial in defining its orientation.

3. Properties:

   • A plane is flat and extends infinitely in two dimensions.
   • Any two distinct planes can either be parallel, intersect in a line, or coincide.
   • The intersection of a line and a plane can be a single point, a line (if the line lies in the plane), or null (if the line is parallel and separate from the plane).

4. Distance from a Point to a Plane:

   • The distance $d$ from a point $(x_0, y_0, z_0)$ to a plane $Ax + By + Cz + D = 0$ is given by: $$d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$$
   • ​∣Ax0​+By0​+Cz0​+D∣​

Relationship between Lines and Planes

• Intersection: A line can either lie on a plane, be parallel to it and not intersect it, or intersect it at exactly one point.
• Parallelism: A line and a plane are parallel if they do not intersect and the line is parallel to a vector lying on the plane.
• Angle: The angle between a line and a plane can be found using the dot product of the line's direction vector and the plane's normal vector.

Understanding lines and planes is fundamental in geometry, algebra, calculus, and various applications across physics, engineering, and computer graphics.