Introduction to Three Dimensional Geometry

Section Formula in 3D

Does it ever come to your mind that how the pilots know where to take the flights? How does he get to know about the position? How do the scientists know where to send the satellites and other things like these? While playing how you can position the ball in the sky or a tossed coin or just a balloon in the air? All these can be done by using Coordinate Geometry and the Section Formulas in 3D.

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Coordinates in 3D

Just like in two dimensions, you can give coordinates to a point in 3D. The three dimensions are the x-axis, y-axis, and z-axis. A point in three dimensions will have coordinates (x, y, z). This ordered tuple has its application in our day to day life.

Take an example of the cricket match. The ball in the sky can be positioned by its coordinates. The position of the submarines, airplanes etc. are all given by its coordinates in 3D.

Section Formulas in 3D

Suppose you draw a line in a white sheet and you put a point anywhere either on the line or not on the line. This point cuts the line segment into two parts. These parts are maybe in some ratio with each other.

By using this ratio we can find the coordinates of the point in a plane in three dimensions. Section formulas in 3D is similar to the section formulas in two dimensions.

Section Formula (Dividing Internally)

Here we will understand how to find the section formula or the coordinates of a point lying on a line. Let AB be a line segment with coordinates (x1, y1, z1) & (x2, y2, z2) respectively. A point C (x, y, z)  divides PR in the ratio m:n internally.

Section Formulas

AL, CN, BM be the perpendiculars on XY-plane. These perpendiculars are parallel with each other. Let L, N, M form collinear points of the line LM. From the point C, we draw a line PR || LM. It is notable that PL intersect the line AL externally and RM intersect BM internally. The triangles APC and BRC are similar. Therefore,

m ⁄ n = AC ⁄ BC = PA ⁄ BR = (PL − AL) ⁄ (BM – RM) = (CN – AL) ⁄ (BM – CN) = (z – z1) ⁄ (z2 – z)

Solving both sides, we have z = (mz2 + nz1) ⁄ (m+n). Similarly, we have y = (my2 + ny1) ⁄ (m+n) and x = (mx2 + nx1) ⁄ (m+n). The coordinates of the point C which divides AB internally at the ratio m:n are

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

Section Formula (Dividing Externally)

Let AB be a line segment with coordinates (x1, y1, z1) & (x2, y2, z2) respectively. A point C (x, y, z) divides PR in the ratio m:n externally. The coordinates of C are given by replacing n by −n:

((mx2 − nx1) ⁄ (m − n), (my2 − ny1) ⁄ (m − n), (mz2 − nz1) ⁄ (m − n)).

Solved Examples for You

Problem: What are the coordinates of a point which divides a line segment into equal halves?

Solution: Let AB be a line segment such that A (x1, y1, z1) and B (x2, y2, z2). C is the mid-point of the line segment AB.

Section Formulas

Here, m = n = 1. Putting in the section formula we have, C (x, y); x = (x1 + x2) ⁄ 2, y = (y1 + y2) ⁄ 2.

Problem: Use the section formula in 3D and find the coordinates of the points L and M which divides the line segment AB internally and externally both at the ratio 3:5? The coordinates of A are (2, 4, 1) and B are (1, 3, 2).

Solution: The coordinates of L (dividing AB internally), x = (3×1 + 5×2) ⁄ (3 + 5) = (3 + 10) ⁄ 8 = 13⁄8, y = (3×3 + 5×4) ⁄ (3 + 5) = (9 + 20) ⁄ 8 = 29⁄8, z = (3×2 + 5×1) ⁄ (3 + 5) = (6 + 5) ⁄ 8 = 11⁄8. The coordinates are (13⁄8, 29⁄8, 11⁄8).

Coordinates of L (dividing AB externally), x = (3×1 − 5×2) ⁄ (3 − 5) = (3 − 10) ⁄ −2 = 7⁄2, y = (3×3 − 5×4) ⁄ (3 − 5) = (9 − 20) ⁄ −2 = 11⁄2, z = (3×2 − 5×1) ⁄ (3 − 5) = (6 − 5) ⁄ −2 = −1⁄2. The coordinates are (7⁄2, 11⁄2, −1⁄2).

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