Suppose there is a point P(x0, y0, z0) and a straight line L. The straight line L can be expressed as point Q(x 1, y 1, z 1) plus a direction vector D(a, b, c). Our goal is to find the distance from point P to line L.
First, we can connect a point Q and a point P on a straight line into a vector QP, which is expressed as vector V (x0-x 1, y0-y 1, z0-z 1).
Then, we calculate the projection of the vector QP on the direction vector D of the straight line L. The projection vector is denoted as the vector proj, its size is denoted as |Proj|, and the direction vector is denoted as the vector n(a, b, c). According to the definition of vector projection, the following relationship can be obtained: Proj = |v| * cosθ, where θ is the included angle between vector v and vector d.
Since vector D is the direction vector of straight line L, and vector V is perpendicular to vector D, that is, vector V is orthogonal to vector N, we can use the inner product of vectors to calculate the relationship between vector V and vector N: v n = | v | | n | * cos (90) = 0.
Substituting vector v and vector n into the above formula, we can get the following relationship: (x0-x1) * a+(y0-y1) * b+(z0-z1) * c = 0.
Further sorting, we can get the formula of the distance from point P to line L:
d = |(x0-x 1)* a+(y0-y 1)* b+(z0-z 1)* c |/√(a^2+b^2+c^2)
Where (x0, y0, z0) is the coordinate of point P, (x 1, y 1, z 1) is the coordinate of point Q on the straight line L, and (a, b, c) is the direction vector of the straight line L. ..
This is the distance formula from point P to straight line L. By calculating this formula, we can solve the distance from any point to straight line in space.
It should be noted that in practical application, we may encounter straight lines expressed by parametric equations or general equations. According to the different expressions of the equation, we need to make corresponding transformations before we can apply the above formula to solve the distance.
I hope the above analysis will help you! If you have any other questions, please feel free to ask.