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Distance Between the Two Points

Let OX, OY and OZ form the rectangular coordinate system. Let and be two points in space. To find the distance between them, first draw planes parallel to the coordinate planes so as to form a rectangular parallelopiped with one diagonal PQ.

Since Δ PQR is right angled at R, we have PQ2 = PR2 + RQ2.
Also Δ PSR is right angled at S and so PR2 = PS2 + SR2.
PQ2 =PS2 + SR2 + RQ2

But we know that
Hence
Therefore .
This gives us the distance between two points.

Corollary


If P is the origin (0, 0, 0), then
Thus the distance between the origin and any point is given by

Example 1
Find the distance between the points

Solution:

The distance between the points is 5 units

Example 2
Show that the points and are collinear

Solution:

We see that
A, B and C are collinear

Example 3
Determine the points in the YZ - plane which is equidistant from the points

Solution:
Let be a point in the YZ - plane. Given PA = PB = PC
Now

Now

Solving (1) and (2) we get

Substituting the value of in (2) we get

The point on the YZ - plane which is equidistant from A, B and C is

Example 4
Prove that the triangle by joining the three points (2,3,4), (3, 4, 2) and (4,2,3) is an equilateral triangle

Solution:
Let A, B and C be the points (2,3,4), (3, 4, 2) and (4,2,3) respectively. Then by the distance formula


We see that AB = BC = AC
The triangle ABC is an equilateral triangle

Example 5
Three vertices of a parallelogram ABCD are . Find the coordinates of the fourth vertex

Solution:
we know that in a parallelogram the diagonals bisect each other

mid point of AC = midpoint of BD
Now mid point of AC =
                             = (1, 0, 2)
Let D be the point then midpoint of
Equating the corresponding coordinates we get

The fourth vertex of the parallelogram is

Example 6
If the origin is the centroid of the triangle PQR with vertices , then find the values of

Solution:
The centroid of the vertices with sides is given by
The centroid of the triangle PQR is

But given the centroid is the origin (0, 0, 0) equating the coordinates to zero we get


Example 7
Find the equation of the set of points P such that 3AP = 2PB where A and B are the points respectively

Solution:
Let P be the points Given 3AP = 2PB





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