3D Coordinate Geometry

In three-dimensional space we can label where any object is using the x-y-z coordinate system
In the 3D cartesian system, the x- and y- axes usually represent lateral space (length and width) and the z-axis represents vertical height

If we have two points with coordinates (x₁ , y₁ , z₁) and (x₂ , y₂ , z₂) then we should be able to find:
- The midpoint of the two points
- The distance between the two points
If the coordinates are labelled A and B then the line segment between them is written with the notation [AB]

The midpoint is the average (middle) point
- It can be found by finding the middle of the x-coordinates and the middle of the y-coordinates
The coordinates of the midpoint will be

$(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}, \frac{z_{1} + z_{2}}{2})$

The distance between two points with coordinates ((x₁ , y₁ , z₁) and (x₂ , y₂ , z₂) can be found using the formula

$d = \sqrt{{(x_{1} - x_{2})}^{2} + {(y_{1} - y_{2})}^{2} + {(z_{1} - z_{2})}^{2}}$

The points A and B have coordinates (-2, 1, 5) and (4, -3, 2) respectively.

Calculate the distance of the line segment AB.

3-2-1-ai-sl-3d-coord-geo-we-solution-i

ii)

Find the midpoint of [AB ].

3-2-1-ai-sl-3d-coord-geo-we-solution-ii

3D Coordinate Geometry (DP IB Maths: AI SL)