Section / ratio – 3 dimensions Calculator

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

(mx₂-nx₁/m-n, my₂-ny₁/m-n, mz₂-nz₁/m-n)

Ratio Internally

(m(x₂) +n(x₁)/m+n, m(y₂) +n(y₁)/m+n, m(z₂) +n(z₁)/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.

x₁ = 1, y₁ = 2, z₁ = 3 and

x₂ = 2, y₂ = 1, z₂ = 3

m = 1, n = 2

Step 2:

The formula for 3-dimensional section/ratio.

(mx₂+nx₁/m+n, my₂+ny₁/m+n, mz₂+nz₁/m+n)

Step 3:

Use the above formula and simplify.

x₁ = 1, y₁ = 2, z₁ = 3 and x₂ = 2, y₂ = 1, z₂ = 3, m = 1, n = 2

(mx₂+nx₁/m+n, my₂+ny₁/m+n, mz₂+nz₁/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.

x₁ = 1, y₁ = 3, z₁ = 5 and

x₂ = 2, y₂ = 3, z₂ = 4

m = 2, n = 3

Step 2:

The formula for 3-dimensional section/ratio.

 (mx₂-nx₁/m-n, my₂-ny₁/m-n, mz₂-nz₁/m-n)

Step 3:

Use the above formula and simplify.

x₁ = 1, y₁ = 3, z₁ = 5 and x₂ = 2, y₂ = 3, z₂ = 4, m = 2, n = 3

= (mx₂-nx₁/m-n, my₂-ny₁/m-n, mz₂-nz₁/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)