How does using cross product to find shortest distance work?

In summary, the method for finding the shortest distance between two skew, non-intersecting lines is to first find the common normal using the formula $ \vec{n} = \frac{\vec{v_1} \times \vec{v_2}}{|\vec{v_1} \times \vec{v_2}|} $. Then, the shortest distance (d) can be obtained by projecting any distance vector onto this normal. The angles between any position vector and the skew lines effectively minimize the projection. The cross product of two coplanar vectors is perpendicular to the plane those vectors are in, and the same applies to two non-coplanar vectors. In this case, the vectors can be "moved"
  • #1
ognik
643
2
A method for finding the shortest distance between 2 skew, non intersecting lines is to 1st find the common normal, using $ \vec{n} = \frac{\vec{v_1} \times \vec{v_2}}{|\vec{v_1} \times \vec{v_2}|} $ I'm looking for a proof or intuition as to why this is true please?

Then apparently we get the shortest distance (d) by projecting any distance vector onto this normal? A distance vector here is any point on one vector minus any point on the other vector. I can obviously choose points to get distance vectors of any magnitude. So my 2nd question is, I assume the angles between any position vector and the skew lines effectively minimise the projection?

But I did a sketch - View attachment 4918 - which doesn't convince me at all about the above ..., here $|\vec{P_2P_1}| \gt |d|$, multiplying that by $Cos \theta $ produces the line from $P_2 \perp \vec{v_1}$ - still > d?
 

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  • #2
What I'm not quite sure of is - I know the cross product of 2 coplaner vectors is perpendicular to the plane those vectors are in, but why is that the same for 2 non-coplaner vectors?
 
  • #3
ognik said:
What I'm not quite sure of is - I know the cross product of 2 coplaner vectors is perpendicular to the plane those vectors are in, but why is that the same for 2 non-coplaner vectors?
The two vectors (unless they are colinear) form the basis vectors for a plane. Two distinct vectors are always coplanar.

-Dan
 
  • #4
Thanks - I just can't visualise it in this case - where the vectors are skew and don't intersect ... the only surface I can visualise in this case is not flat?
 
  • #5
ognik said:
Thanks - I just can't visualise it in this case - where the vectors are skew and don't intersect ... the only surface I can visualise in this case is not flat?
We can always "move" the vectors so that they meet end to end.

-Dan
 
  • #6
topsquark said:
We can always "move" the vectors so that they meet end to end.

-Dan
Hi - after much thought I decided we must be talking about the directions of the vectors? IE there is a vector parallel to one of the vectors, which does intersect the other vector. Then the cross product makes sense as being perpendicular to both directions?
 
  • #7
ognik said:
Hi - after much thought I decided we must be talking about the directions of the vectors? IE there is a vector parallel to one of the vectors, which does intersect the other vector. Then the cross product makes sense as being perpendicular to both directions?
In a word: Yup! (Sun)

-Dan
 

FAQ: How does using cross product to find shortest distance work?

How is the cross product used to find the shortest distance between two points?

The cross product is a mathematical operation that results in a vector perpendicular to two given vectors. This perpendicular vector can be used to represent the shortest distance between two points in 3-dimensional space.

What are the steps involved in using cross product to find the shortest distance?

The first step is to find the cross product of the two vectors connecting the two points. Then, calculate the magnitude of the resulting vector. Finally, divide the magnitude by the magnitude of the second vector to get the shortest distance between the two points.

Why is cross product used instead of other methods to find the shortest distance?

Cross product is used because it provides a direct and efficient way to calculate the shortest distance between two points in 3-dimensional space. Other methods may involve more complex calculations and may not always provide accurate results.

Can cross product be used to find the shortest distance between points in any dimension?

No, cross product is only applicable in 3-dimensional space. In higher dimensions, other methods such as dot product or scalar projection may be used to find the shortest distance between points.

Are there any limitations to using cross product to find the shortest distance?

One limitation is that cross product can only be used when the two points are not collinear. If the points are collinear, then the shortest distance between them would be the length of the connecting vector. Additionally, cross product may not always provide the shortest distance if the points are not within the same plane.

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