IBMaths: AI HLDPRevision Notes3. Geometry & Trigonometry3.2 Geometry of 3D Shapes3.2.1 3D Coordinate Geometry

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

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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

begin mathsize 22px style stretchy left parenthesis fraction numerator x subscript 1 plus x subscript 2 over denominator 2 end fraction space comma space fraction numerator space y subscript 1 plus y subscript 2 over denominator 2 end fraction space comma space fraction numerator z subscript 1 plus z subscript 2 over denominator 2 end fraction stretchy right parenthesis end style

    • 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

begin mathsize 22px style d equals blank square root of open parentheses x subscript 1 minus blank x subscript 2 close parentheses squared space plus space open parentheses y subscript 1 minus blank y subscript 2 close parentheses squared space plus space open parentheses z subscript 1 minus blank z subscript 2 close parentheses squared end root end style

    • 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.

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

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    Amber

    Author: Amber

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    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.