Are there skew lines in more than 3 dimensions?
By James Williams
In our environment, two straight lines are, in general, skew lines.
The term skew means that the straight lines are not parallel, do not cross, and most importantly, that there is a shortest distance between them. The centre of this connection can be called the midpoint of the skew lines.
If we go to more dimensions, such as four or five, this yields a number of questions:
- Do skew lines exist? - Probably yes.
- Is there a unique shortest distance perpendicular to both lines? - This is harder.
- Is there a midpoint? - This depends on (2).
- Does the answer depend on whether the number of dimensions is odd or even?
- Does something special happen in 7 dimensions, like for the vector product?
And a new, extra question:
- In 3 dimensions, skew lines hinder each other in their motion. Is this also the case in higher dimensions?
This does not seem easy to answer. Thank you for any help!
$\endgroup$ 51 Answer
$\begingroup$Skew lines certainly exist in higher dimensions. As pointed out in the comments, for a pair of lines in $\mathbb R^n$ with $n\ge3$ you can always find a three-dimensional affine subspace that contains the two lines. Informally, there are even more ways for two lines to “miss” each other as you add dimensions.
The rest pretty much follows from the above observation. Alternatively, if you parameterize the lines as $\mathbf p_1+s\mathbf v_1$ and $\mathbf p_2+t\mathbf v_2$, respectively, observe that the square of the distance between these points is a quadratic function of $s$ and $t$ regardless of the dimensionality of the ambient space.
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