Question

What is the symmetric equation of a line in three-dimensional space?

Answer 1

The symmetric equation of the line with the direction vector `vec{v}=(a,b,c)` passing through the point `(x_0,y_0,z_0)` is:

`{x-x_0}/a={y-y_0}/b={z-z_0}/c`,

where none of `a,b` and `c` are zero.

If one of `a,b`, and `c` is zero; for example, `c=0`, then we can write:

`{x-x_0}/a={y-y_0}/b` and `z=z_0`.

If two pf `a,b`, and `c` are zero; for example, `b=c=0`, then we can write:

`y=y_0` and `z=z_0`

(There is no restriction on `x`, it can be any real number. )

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I hope that this was helpful.