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# Section / ratio – 3 dimensions Calculator

Insert the given values and calculate Section/ratio – 3 dimensions by using this calculator internally and externally.

Table of Contents:

## Section/ratio – 3 dimensions

A section or cross-section is the shape that results from cutting a three-dimensional object with a plane in mathematics. This cut produces a two-dimensional form that symbolizes the point where the item and the plane intersect.

**Ratio Externally**

(mx_{2}-nx_{1}/m-n, my_{2}-ny_{1}/m-n, mz_{2}-nz_{1}/m-n)

**Ratio Internally**

(m(x_{2}) +n(x_{1})/m+n, m(y_{2}) +n(y_{1})/m+n, m(z_{2}) +n(z_{1})/m+n)

## Solved Examples

**Example 1:**

Evaluate the coordinates of the point divided by the line joining with two points (1, 2, 3) and (1, 2, 3) by the ratio of 1:2 internally.

**Solution **

**Step 1:**

Write the data.

x_{1} = 1, y_{1} = 2, z_{1} = 3 and

x_{2} = 2, y_{2} = 1, z_{2} = 3

m = 1, n = 2

**Step 2:**

The formula for 3-dimensional section/ratio.

(mx_{2}+nx_{1}/m+n, my_{2}+ny_{1}/m+n, mz_{2}+nz_{1}/m+n)

**Step 3:**

Use the above formula and simplify.

x_{1} = 1, y_{1} = 2, z_{1} = 3 and x_{2} = 2, y_{2} = 1, z_{2} = 3, m = 1, n = 2

(mx_{2}+nx_{1}/m+n, my_{2}+ny_{1}/m+n, mz_{2}+nz_{1}/m+n)

= (1*2+2*1/1+2, 1*1+2*2/1+2, 1*3+2*3/1+2)

= (2+2/3, 1+4/3, 3+6/3)

= (4/3, 5/3, 9/3)

= (1.3, 1.6, 3)

**Example 2:**

Evaluate the coordinates of the point divided by the line joining with two points (1, 3, 5) and (2, 3, 4) by the ratio of 2:3 externally.

**Solution **

**Step 1:**

Write the data.

x_{1} = 1, y_{1} = 3, z_{1} = 5 and

x_{2} = 2, y_{2} = 3, z_{2} = 4

m = 2, n = 3

**Step 2:**

The formula for 3-dimensional section/ratio.

(mx_{2}-nx_{1}/m-n, my_{2}-ny_{1}/m-n, mz_{2}-nz_{1}/m-n)

**Step 3:**

Use the above formula and simplify.

x_{1} = 1, y_{1} = 3, z_{1} = 5 and x_{2} = 2, y_{2} = 3, z_{2} = 4, m = 2, n = 3

= (mx_{2}-nx_{1}/m-n, my_{2}-ny_{1}/m-n, mz_{2}-nz_{1}/m-n)

= (2*2-3*1/2-3, 2*3-3*3/2-3, 2*4-3*5/2-3)

= (4-3/-1, 6-9/-1, 8-15/-1)

= (1/-1, -3/-1, -7/-1)

= (-1, 3, 7)