Three-dimensional geometry
Distance, midpoints and direction ratios in 3D.
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In 3D, a point needs three numbers: across, along and up.
Look: P(2, 3, 4): 2 along x, 3 along y, then 4 up.
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1Distance from (0, 0, 0) to (1, 2, 2)?
2Middle of (2, 4, 6) and (4, 0, −2)?
3Points with z = 0 lie on which flat plane?
! Careful!
✗Using only x and y.
✓Include z too.
✗Forgetting the square root.
✓Last step: √.
See the whole lesson on one page
In 3D, a point needs three numbers: across, along and up.
A point is (x, y, z). Distance works like Pythagoras, with one more square under the root.
Think of it like thisIn a mall: which shop column, which row, and which floor. Three numbers find your spot.
Solved together
How far is (2, 3, 6) from the origin (0, 0, 0)?
Square each number and add.
4 + 9 + 36 = 49Square root.
√49 = 7
Answer7 units.
For older learners and grown-ups: a little more
A line through point A with direction b: r = a + λb. Direction cosines are direction ratios divided by length; their squares add to 1.
Now practise it
Kanakku makes new questions on three-dimensional geometry every time. It shows the working and gives a hint when you are stuck.