Introduction
Presence of material objects in relation to one another or their movement makes us feel about what is ordinarily known as space. In other words, the notion of space is closely associated with material objects in relative rest and/or motion. We often regard a point as an idealized model of a small material object (or particle).
Small material objects or particles, when taken together, form bigger objects. Just the same way, we agree that points, when taken together, give rise to various forms or shapes. A space may therefore be visualized as the set of all points that give rise to various forms or shapes.
A systematic study of such set of points or shapes can be done by associating each of such points with an ordered set of real numbers. Such numbers are said to be coordinates of a given point in the space under consideration.
Points in Space
We began our study of coordinate geometry by establishing a one-to-one correspondence between the points on a line and the real numbers. This was followed by associating each point in a plane with an ordered pair of real numbers. Now we intend to explain how each point in the ordinary space (or three-dimensional space) can be associated with an ordered triple of real numbers.
To start with, we consider three mutually perpendicular lines intersecting at a fixed point. We call this point the origin and the three lines the coordinate axes. They are usually labeled as the x- axis, the y-axis and the z-axis.
Exercise – 11.1
Before we move towards the solution of this chapter. Let’s have a short discussion on the topics we have to study in coordinate in space second exercise. We already have discussed what we have to study in first exercise in our previous article. If you haven’t check that out then you should have a look at that before you start second exercise.
Projection
Projections are the transformation of points and lines on one plane onto another plane by connecting corresponding points on the two planes with parallel lines that go across those planes
a) Projection of a point on a line or a plane
The projection of a point P on a line AB or a plane CDEF is the foot L of the perpendicular from the point P on the line AB or plane CDEF. In the first case, it is also interpreted as the A- point L of intersection of the plane through the given point P and perpendicular to the given line.
b) Projection of a line segment on a line
The projection of the line segment AB of a given line on another line CD is the segment A’B’ of CD where A’ and B’ are the projections of A and B on CD. Here we may also say that A’ and B’ are the points in which planes through A and B perpendicular to CD meet the line CD
