The discussion revolves around proving a formula for the distance between a point and a line in vector terms. The original poster presents a specific vector-based expression involving the cross product of two vectors, one from the point to the line and another parallel to the line.
The conversation is ongoing, with participants exploring different interpretations of the vectors and their geometric meanings. Some guidance has been offered regarding the relationship between the vectors and the area they define, but no consensus has been reached on how to proceed with the proof.
There is an emphasis on the need for an algebraic proof, and participants are grappling with the definitions and relationships of the vectors involved, particularly regarding the interpretation of the norm of vector v.
You mean sin θ, right?uart said:Use u x v = |u| |v| cos(theta)
vela said:You mean sin θ, right?
No and no. The norm of a vector v has a perfectly good geometric interpretation - it's the length of the vector. A line has infinite length - maybe you meant line segment?aero_zeppelin said:Ok, so we know that the norm of uxv is the area of the parallelogram formed by u and v...
The norm of v has no geometric interpretation (its just the length of a line)
aero_zeppelin said:How can we relate this so it gives the distance between P and the line...
any more ideas??
Mark44 said:No and no. The norm of a vector v has a perfectly good geometric interpretation - it's the length of the vector. A line has infinite length - maybe you meant line segment?