Distance from a Point to a Line using the Cross Product
In this video I go over a method for determining the shortest distance between a point and a line in 3D by using the length of the cross product and the area of a parallelogram. The position vector between a point and a line forms a parallelogram with the direction vector of that line, the height of which is the perpendicular distance or shortest distance between the point and line. This means that we can use the cross product to determine the area of the parallelogram and divide it by the length of the direction vector to obtain the distance.
Note that I also include the distance formula in 2D from my earlier video, although the same method above will obtain the same result and we just have to set the z coordinates to zero in the 3D version:
Timestamps:
- Question 17: Distances between points, lines, and planes: 0:00
- Solution to (a): Distance from a point to a line: 0:21
- Illustration of the method: 1:30
- Distance is the height of the parallelogram: 4:00
- Distance formula in 2D: 5:20
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