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It is not immediately possible to use the formula for the distance between a point and a line if the line is given in vector form. In the following example, we show how to find the required distance if the line is given in vector form. EXAMPLE 2 Selecting a strategy to determine the distance between a point and a line in Determine the distance ...
Chapter 25: distance from a point to a line. 點到直線距離. Theorem 25.1 {P = P (x0,y0) L =L(x,y) = Ax+By+C= 0,A2+B2 ≠0 ⇓ d(P,L) = |Ax0 +By0+C| √A2+B2 { P = P ( x 0, y 0) L = L ( x, y) = A x + B y + C = 0, A 2 + B 2 ≠ 0 ⇓ d ( P, L) = | A x 0 + B y 0 + C | A 2 + B 2. https://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line.
Example: Find the distance from (i) the point (1;2;4) to the line L through (2;3;2) which is parallel to ( 1; 1;5); (ii) the point (1;1; 2) to the line Lthrough (3; 3;2) which is parallel to (1; 2;2).
The distance between a point and a line, is defined as the shortest distance between a fixed point and any point on the line. It is the length of the line segment that is perpendicular to the line and passes through the point.
Consider a vector that gives you the direction of the line: $\vec{v}=(1,1,1).$ Consider a point of the line: $P(2,2,2).$ Now, the vectors $\vec{v}=(1,1,1)$ and $\vec{PS}=(0,0,-1)$ determine a paralellogram.
This example uses a setup like the one for the distance from a point to a line. Setting s= 0 gives the point P(1,2,5) on the first line. Setting t= 0 gives the point Q(1,0,8) on the second line. Hence, −−→ PQ= (0,−2,3) is a vector from the first line to the second line.
Check whether the lines intersect by setting their parametric equations equal. If they intersect, the distance is zero. If they do not intersect and parallel (these can be observed by comparing the direction vectors), late any point on one line and calculate the distance to another line. If the lines do not intersect and are nor parallel, they ...
