Practice question · Sort into groups
Sort each description by the object it defines.
Groups: Describes a line · Describes a plane
- r = r0 + t d
- (x, y, z) = (1, 2, 3) + t(0, 1, 1)
- The points whose displacement from a fixed point is orthogonal to a fixed vector
- n . (r - r0) = 0
- 2x + 3y - z = 5
Hints
- A single free parameter traces out a line; an orthogonality condition on a normal carves out a plane.
- Count how much freedom each description leaves.
Show the answer
Describes a line: r = r0 + t d, (x, y, z) = (1, 2, 3) + t(0, 1, 1)
Describes a plane: n . (r - r0) = 0, 2x + 3y - z = 5, The points whose displacement from a fixed point is orthogonal to a fixed vector
Why
One parameter t gives one degree of freedom, hence a line. A single dot-product condition removes one degree of freedom from space, leaving two, a plane. The last item is just the geometric description of n . (r - r0) = 0 in words.
Practise Lines and Planes in Space
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