Advanced Mathematics

My goals
By the end of this unit, I will be able to: 

• plot points in three dimensions. 
• find equation of straight lines in three dimensions.  
• find equation of planes in three dimensions.
• position of lines and planes in space. 
• find equation of sphere.
  

Introduction 

In 2-Dimensions, the position of a point is determined by two coordinates x and y. However, in 3-Dimensions the position of point determined by three coordinates x, y, z obtained with reference to three straight lines (x-axis , y-axis and z-axis respectively) intersecting at right angles. 

In the plane, a line is determined by a point and a number giving the slope of the line. However, in 3-dimensional space, a line is determined by a point and a direction given by a parallel vector, called the direction vector of the line. 

In a 2-dimensional coordinate system, there were three possibilities when considering two lines: intersecting lines, parallel lines and the two were actually the same line but in 3-dimensional space. There is one more possibility: Two lines may be skew, which means they do not intersect, but are not parallel. 

In space, a plane is determined by a point and two direction vectors which form a basis (linearly independent vectors).

Advanced Mathematics Learner’s Book Five

Sphere is the locus of a point in space which remains at a constant distance called the radius from a fixed point called the centre of the sphere.

8.1. Points in 3 dimensions

8.1.1. Location of a point in space

Activity 8.1:

Consider the point ()2,3,5A in space, on a piece of paper

1. Copy the following figure

2. From x-coordinate 2, draw a line parallel to y-axis.

3. From y-coordinate 3, draw another line parallel to x-axis.

4. Now you have a point of intersection of two lines, let us call it P. From this point, draw another line parallel to z-axis and another joining this point and origin of coordinates which is line OP.

5. From z-coordinate, draw another line parallel to the line OP.

6. Draw another line parallel to z-axis and passing through point P.

Suppose that we need to represent the point 2,3,5A in space. From Activity 8.1, we have

Let us see it using a box

8.1.2. Coordinates of a midpoint of a segment  and centroid of a geometric figure

8.2. Straight lines in 3 dimensions

8.2.1. Equations of lines

In the plane, a line is determined by a point and a number giving the slope of the line. In 3-dimensional space, a line is determined by a point and a direction given by a parallel vector, called the direction vector of the line. We will denote lines by capital letters such as L, M,...

             

               

            

   

             

         

         

      

       

       

  

One of the methods of finding this shortest distance is to write the parametric form of any point of each given line. Next, find the vector joining the points in parametric form which will be the vector in the direction of the common perpendicular of both lines. Now, the dot product of this vector and the direction vector of each line must be zero. This will help us to find the value of parameters and hence two points (one on the first line and another on the second line). The common perpendicular of the two lines passes through these two points. Then, the distance between these two points is the required shortest distance between the two lines.
Using this method, we can find the equation of the common perpendicular since we have two points where this common perpendicular passes.
Note that if two lines intersect (not skew lines), the shortest distance is zero. 

 

 

A line L is perpendicular to plane α if and only if each direction vector of L is perpendicular to each direction vector of α or the scalar product of direction vector of the line and the direction vector of the plane is zero.
In this case, the direction vector of the line is perpendicular to the plane and is said to be the normal or orthogonal vector of the plane.
Note that the normal vector of the plane can be found by finding the vector product of its two non proportional direction vectors.

Thus, the angle between the given plane and the given line is 67.8 degrees.

b) Angle between two planes

It is important to choose the correct angle here. It is defined as the angle between two lines, one in each plane, so that they are at right angles to the line of intersection of the two planes (like the angle between the tops of the pages of an open book).

When finding the image of a point P with respect to the plane α , we need to find the line, say L, through point P and perpendicular to the plane α . 

The next is to find the intersection of line L and plane α , say N. Now, if Q is the image of P, the point N is the midpoint of PQ. From this, we can find the coordinate of Q.

 Similarly, if we need the image of a line, we will need the parametric form of any point on the line and then find its image using the same method. The image will be in parametric form. 

Now, replacing the parameter by any two chosen values in the obtained image, we will get two points. From these two points, we can find the equations of the line which will be the image of the given line.