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Shortest distance between a Line and a Point in a 3-D plane
• Difficulty Level : Easy
• Last Updated : 08 Feb, 2021

Given a line passing through two points A and B and an arbitrary point C in a 3-D plane, the task is to find the shortest distance between the point C and the line passing through the points A and B.
Examples:

```Input: A = (5, 2, 1), B = (3, 1, -1), C = (0, 2, 3)
Output: Shortest Distance is 5

Input: A = (4, 2, 1), B = (3, 2, 1), C = (0, 2, 0)
Output: Shortest Distance is 1```

Consider a point C and a line that passes through A and B as shown in the below figure. Now Consider the vectors, AB and AC and the shortest distance as CD. The Shortest Distance is always the perpendicular distance. The point D is taken on AB such that CD is perpendicular to AB. Construct BP and CP as shown in the figure to form a Parallelogram. Now C is a vertex of parallelogram ABPC and CD is perpendicular to Side AB. Hence CD is the height of the parallelogram. Note: In the case when D does not fall on line segment AB there will be a point D’ such that PD’ is perpendicular to AB and D’ lies on line segment AB with CD = PD’.
The magnitude of cross product AB and AC gives the Area of the parallelogram. Also, the area of a parallelogram is Base * Height = AB * CD. So,

`CD = |ABxAC| / |AB|`

Below is the CPP program to find the shortest distance:

## CPP

 `// C++ program to find the Shortest``// Distance Between A line and a``// Given point.``#include``using` `namespace` `std;` `class` `Vector {``private``:``    ``int` `x, y, z;``    ``// 3D Coordinates of the Vector` `public``:``    ``Vector(``int` `x, ``int` `y, ``int` `z)``    ``{``        ``// Constructor``        ``this``->x = x;``        ``this``->y = y;``        ``this``->z = z;``    ``}``    ``Vector operator+(Vector v); ``// ADD 2 Vectors``    ``Vector operator-(Vector v); ``// Subtraction``    ``int` `operator^(Vector v); ``// Dot Product``    ``Vector operator*(Vector v); ``// Cross Product``    ``float` `magnitude()``    ``{``        ``return` `sqrt``(``pow``(x, 2) + ``pow``(y, 2) + ``pow``(z, 2));``    ``}``    ``friend` `ostream& operator<<(ostream& out, ``const` `Vector& v);``    ``// To output the Vector``};` `// ADD 2 Vectors``Vector Vector::operator+(Vector v)``{``    ``int` `x1, y1, z1;``    ``x1 = x + v.x;``    ``y1 = y + v.y;``    ``z1 = z + v.z;``    ``return` `Vector(x1, y1, z1);``}` `// Subtract 2 vectors``Vector Vector::operator-(Vector v)``{``    ``int` `x1, y1, z1;``    ``x1 = x - v.x;``    ``y1 = y - v.y;``    ``z1 = z - v.z;``    ``return` `Vector(x1, y1, z1);``}` `// Dot product of 2 vectors``int` `Vector::operator^(Vector v)``{``    ``int` `x1, y1, z1;``    ``x1 = x * v.x;``    ``y1 = y * v.y;``    ``z1 = z * v.z;``    ``return` `(x1 + y1 + z1);``}` `// Cross product of 2 vectors``Vector Vector::operator*(Vector v)``{``    ``int` `x1, y1, z1;``    ``x1 = y * v.z - z * v.y;``    ``y1 = z * v.x - x * v.z;``    ``z1 = x * v.y - y * v.x;``    ``return` `Vector(x1, y1, z1);``}` `// Display Vector``ostream& operator<<(ostream& out,``                    ``const` `Vector& v)``{``    ``out << v.x << ``"i "``;``    ``if` `(v.y >= 0)``        ``out << ``"+ "``;``    ``out << v.y << ``"j "``;``    ``if` `(v.z >= 0)``        ``out << ``"+ "``;``    ``out << v.z << ``"k"` `<< endl;``    ``return` `out;``}` `// calculate shortest dist. from point to line``float` `shortDistance(Vector line_point1, Vector line_point2,``                    ``Vector point)``{``    ``Vector AB = line_point2 - line_point1;``    ``Vector AC = point - line_point1;``    ``float` `area = Vector(AB * AC).magnitude();``    ``float` `CD = area / AB.magnitude();``    ``return` `CD;``}` `// Driver program``int` `main()``{``    ``// Taking point C as (2, 2, 2)``    ``// Line Passes through A(4, 2, 1)``    ``// and B(8, 4, 2).``    ``Vector line_point1(4, 2, 1), line_point2(8, 4, 2);``    ``Vector point(2, 2, 2);` `    ``cout << ``"Shortest Distance is : "``         ``<< shortDistance(line_point1, line_point2, point);` `  ``return` `0;``}`
Output:
`Shortest Distance is : 1.63299`

Time Complexity: O(1)

Auxiliary Space: O(1)

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