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3D Coordinate Geometry (DP IB Maths: AI HL)
Revision Note
3D Coordinate Geometry
How does the 3D coordinate system work?
- 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
What can we do with 3D coordinates?
- 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]
How do I find the midpoint of two points in 3D?
- 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
-
- This is given in the formula booklet, you do not need to remember it
How do I find the distance between two points in 3D?
- The distance between two points with coordinates ((x1 , y1 , z1) and (x2 , y2 , z2) can be found using the formula
-
- This is given in the formula booklet, you do not need to remember it
Worked example
The points A and B have coordinates (-2, 1, 5) and (4, -3, 2) respectively.
i)
Calculate the distance of the line segment AB.
ii)
Find the midpoint of [AB ].
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