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Coordinates of Points Externally/Internally(3 dimension) Calculator

Enter the values of Points(three dimensional coordinates) and ratio

Calculator

(Coordinates)

Ratio
mn

 

Calculator find the coordinates of point p which divides the line joining two Entered points A and B internally or externally, in a specified ratio m and n. Coordinates of point is a set of values that is used to determine the position of a point in a three dimensional Cartesian coordinate.

Understanding the Section Formula in 3D Space

The Section Formula is a fundamental tool in coordinate geometry used to find the exact position of a point that divides a line segment. While simple in 2D, in 3D space, we account for the depth (Z-axis), which is vital for engineering, computer graphics, and physics.

A (x1, y1, z1) B (x2, y2, z2) P (Moving Ratio) Visualizing Ratio m : n

Fig: Point P dividing segment AB. Note how the ratio changes as P moves.

Internally divided line segment:

The given three dimensional coordinates forms a line AB where point P(xp, yp, yp) lies outside of the line segment AB. It can divide the AB externally as showing below:

Coordinates of Points Internally(3 dimension)

Formula (internally division of a line segment)

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

Externally divided line segment:

The given 3D coordinates forms a line AB where point P(xp, yp, zp) lies on the line segment AB joining Point P. It is showing bellow:

Coordinates of Points Externally(3 dimension)

Formula (externally division of a line segment)

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

Internal vs. External Division

Depending on where the point P is located relative to the endpoints A and B, the division is classified into two types:

  • Internal Division: The point $P$ lies between $A$ and $B$. This is commonly used to find the center of mass or midpoint.
  • External Division: The point $P$ lies outside the segment $AB$, but on the same extended line. This is used in trajectory calculations and vector extensions.

How to Calculate Manually

To find the coordinates $(x, y, z)$ of point $P$, we use the weighted average of the coordinates of $A$ and $B$ based on the ratio $m:n$.

Pro Tip: When the ratio $m:n$ is $1:1$, the section formula simplifies to the Midpoint Formula: $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2})$.