Quote: "A journey of 1000 miles begins with a single step"
Point
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A location in a plane or in space, having zero dimensions.
Lines
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A line is made up of an infinite number of points. Points in the same line are called colinear. Between each point is another point. This continues on forever. Thus a line is a one-dimensional figure. That is, a line has length, but no width or height.
A line can be defined by two points A and B. its in infinite length, passing through the points and extending forever in both directions. A line segment between two points A and B, the line that contains them is the set of points consisting of the distinct points A and B, all of the points between them.But the line segment AB extends from to A to B only. A ray is semi infinite. It is specified by a point and direction through its end points. A ray that contains two points A and B is the set of points consisting of the distinct points A and B,all of the points between them, and all points for which B is between them and A.
Every line is either horizontal, vertical or oblique. Horizontal lines have a slope of zero. Vertical lines are said to have infinite slope. In space, vertical lines never meet, but it is possible for horizontal lines to meet. There are four different relationships that two lines can have.
Lines can be identical, intersecting, parallel, perpendicular, or skew. Identical lines are lines that coincide. Therefore, they are the same line. The second one is the most obvious. Intersecting lines are lines that share a point. Parallel lines are coplanar lines that never intersect. They always have a certain distance between them and always have the same direction. Perpendicular lines are lines that intersect in one point and form a 90 degree angle. Skew lines only happen in space. They are noncoplanar lines that never intersect. Unlike parallel lines, however, they don't always have a set distance between them, nor do they always have the same direction.
Planes
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Planes are two-dimensional. A plane has length and width, but no height, and extends infinitely on all sides.Planes are thought of as flat surfaces. A plane is made up of an infinite amount of lines. Some properties of planes:-
1)Any line joining two points on a plane lies wholly in that plane.
2)One and only one plane can be drawn through three non-collinear points
3)One and only one plane can be drawn to contain two concurrent lines.
4)One and only one plane can be drawn perpendicular to a given direction and at a given distance from the origin
5)One and only one plane can be drawn through a given point and perpendicular to a given direction.
Two-dimensional figures are called plane figures. e.g. rectangle, circle, pentagons etc.A closed plane figure made up of several line segments that are joined together is called a polygon.
Space
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Space is the set of all points. It is made up of an infinite number of planes.
Figures in space are called solids or surfaces. eg. cube, sphere etc.
A three-dimensional solid that is bounded by plane polygons is called a polyhedron.
Locus
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Locus of a moving point is the path traced by the point satisfying some geometrical
conditions.
Change of axes
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Translation: To alter the origin of coordinates without altering the direction of axes.
Rotation : To change the direction of axes of coordinates without changing the origin.
Conic Sections
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The curves obtained by slicing a cone with a plane that does not pass through the vertex are called conic sections
Parametric Representation
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Suppose we have a point moving around in the (x, y) plane. At each time t the particle will be at some point whose coordinates we can write as (x(t), y(t)). i.e. the x and y coordinates of the point are given as functions of the parameter t. Many curves can be most conveniently expressed in the form x = x(t) y = y(t) where t is some parameter. t need not be time, though it often helps to think about it in that way. This is called a parametric representation of the curve. e.g. Parametric eqn of a circle x2 +y2 = r2 is x = r cos(t), y = r sin( t)
