Analytical Geometry:
Another term for analytical geometry is Cartesian geometry or co ordinate geometry. It refers to the study of relationships between points, lines, planes etc against a back drop of the co ordinate system, be it in three dimensions or two dimensions. In our lower grade normal geometry, we knew of geometric shapes such as triangles, angles, squares, rectangles etc. The same shapes in co ordinate geometry are described by the co ordinates of their vertices, or by the equations and slopes of the lines joining these vertices.
The basic of analytical geometry deals with the concept of lines. The topics covered are parallel lines, perpendicular lines and inclination of lines with respect to the coordinate systems or with respect to each other. A line is described by its slope or gradient. This slope can be found using the co ordinates of two points that lie on the line. If two lines have the same gradient then they are said to be parallel to each other. If the product of the slopes of two lines is -1, then the two lines are said to be perpendicular to each other. The slope can also be defined by the tangent of the angle that the line makes with the positive x axis direction. Thus if we know the slope we can use the arctan function to find the inclination of the line with respect to the positive x direction.
All the other geometric shapes are studies with the help of this concept of straight lines. For example consider a quadrilateral. It is a closed shape formed by four line segments that are parts of four lines. If both the pairs of opposite sides are parallel and congruent to each other, (that means if both the pairs of opposite sides have the same slope and same length) then such a quadrilateral is called a parallelogram. In a parallelogram if all the adjacent sides are perpendicular to each other, (in other words, if the product of the slopes of adjacent sides is -1), then the parallelogram becomes a rectangle. Similarly other geometric shapes can also be studied in this way.
Analytical geometry is most useful in studying three dimensional objects. This is sometimes also called analytical solid geometry. In three dimnesional space, there are three coordinate systems that can be used. Besides the Cartesian co ordinate system, there can also be the cylindrical coordinate system and the spherical co ordinate system. All these three systems can be interchangeably used to study various types of curves, surfaces or solids in space. Just like how a point in the Cartesian co ordinate system is defined by its x y and z co ordinate, in the cylindrical co ordinate system it is defined by three parameters. They are, (a) its distance from the origin (r), (b) the angle of the line joining the point and the origin with the positive x axis (θ) and (c) its perpendicular distance from the x-y plane (z co ordinate).
Also check out the video streaming of Analytical Geometry
Another term for analytical geometry is Cartesian geometry or co ordinate geometry. It refers to the study of relationships between points, lines, planes etc against a back drop of the co ordinate system, be it in three dimensions or two dimensions. In our lower grade normal geometry, we knew of geometric shapes such as triangles, angles, squares, rectangles etc. The same shapes in co ordinate geometry are described by the co ordinates of their vertices, or by the equations and slopes of the lines joining these vertices.
The basic of analytical geometry deals with the concept of lines. The topics covered are parallel lines, perpendicular lines and inclination of lines with respect to the coordinate systems or with respect to each other. A line is described by its slope or gradient. This slope can be found using the co ordinates of two points that lie on the line. If two lines have the same gradient then they are said to be parallel to each other. If the product of the slopes of two lines is -1, then the two lines are said to be perpendicular to each other. The slope can also be defined by the tangent of the angle that the line makes with the positive x axis direction. Thus if we know the slope we can use the arctan function to find the inclination of the line with respect to the positive x direction.
All the other geometric shapes are studies with the help of this concept of straight lines. For example consider a quadrilateral. It is a closed shape formed by four line segments that are parts of four lines. If both the pairs of opposite sides are parallel and congruent to each other, (that means if both the pairs of opposite sides have the same slope and same length) then such a quadrilateral is called a parallelogram. In a parallelogram if all the adjacent sides are perpendicular to each other, (in other words, if the product of the slopes of adjacent sides is -1), then the parallelogram becomes a rectangle. Similarly other geometric shapes can also be studied in this way.
Analytical geometry is most useful in studying three dimensional objects. This is sometimes also called analytical solid geometry. In three dimnesional space, there are three coordinate systems that can be used. Besides the Cartesian co ordinate system, there can also be the cylindrical coordinate system and the spherical co ordinate system. All these three systems can be interchangeably used to study various types of curves, surfaces or solids in space. Just like how a point in the Cartesian co ordinate system is defined by its x y and z co ordinate, in the cylindrical co ordinate system it is defined by three parameters. They are, (a) its distance from the origin (r), (b) the angle of the line joining the point and the origin with the positive x axis (θ) and (c) its perpendicular distance from the x-y plane (z co ordinate).
Also check out the video streaming of Analytical Geometry
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