 

Distance from a point to a line  3Dimensional.Page navigation:
Definition. Distance from a point to a line — is equal to length of the perpendicular distance from the point to the line.Distance from a point to a line in space formulaIf M_{0}(x_{0}, y_{0}, z_{0}) point coordinates, s = {m; n; p} directing vector of line l, M_{1}(x_{1}, y_{1}, z_{1})  coordinates of point on line l, then distance between point M_{0}(x_{0}, y_{0}, z_{0}) and line l can be found using the following formula:
Proof of the formula of distance from a point to a line for the space problemIf l is line equation then s = {m; n; p} is directing vector of line, M_{1}(x_{1}, y_{1}, z_{1})is coordinates of point on line. From properties of cross product it is known that the module of cross product of vectors is equal to the area of a parallelogramme constructed on these vectors A = M_{0}M_{1}×s. On the other hand parallelogramme area is equal to product of its side on height spent to this side A = sd. Having equated the areas it is simple to receive the formula of distance from a point to a line. Examples of tasks with from a point to a line in spaceExample 1. To find distance between point M(0, 2, 3) and line
Solution. From line equation find:
s = {2; 1; 2}  directing vector of line; Then M_{0}M_{1} = {3  0; 1  2; 1  3} = {3; 1; 4}
= i ((1)·2  (4)·1)  j (3·2  (4)·2) + k (3·1 (1)·2) = {2; 14; 5}
Answer: distance from point to line is equal to 5. Analytic geometry: Introduction and contentsDistance between two pointsMidpoint. Coordinates of midpointEquation of a lineEquation of a planeDistance from point to planeDistance between two planesDistance from a point to a line  2DimensionalDistance from a point to a line  3DimensionalAngle between two planesAngle between line and plane
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