# 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 Î” PQR is right angled at R, we have PQ

^{2}= PR

^{2}+ RQ

^{2}.

Also Î” PSR is right angled at S and so PR

^{2 }= PS

^{2}+ SR

^{2}.

âˆ´ PQ

^{2}=PS

^{2}+ SR

^{2 }+ RQ

^{2}

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:**

âˆ´ The distance between the points is 5 units

**Example 2**

Show that the points and are collinear

**Solution:**

We see that

âˆ´ 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*

âˆ´ 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

âˆ´ 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

âˆ´ 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

âˆ´ 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

âˆ´ 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