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  Distance Formula Between Two Parallel Lines Calculating the distance between two parallel lines is a common problem in geometry. Parallel lines are lines in the same plane that never intersect and always have the same slope. The shortest distance between two parallel lines is the perpendicular distance from any point on one line to the other line. This distance is constant along the entire length of the parallel lines. Equation of a Line in 2D In a two-dimensional (2D) Cartesian coordinate system, the equation of a straight line can be written in the general form: Ax +By +C=0Ax + By + C = 0Where: AA and BB are the coefficients that define the slope of the line, CC is the constant that determines the position of the line relative to the origin. For two parallel lines, their slopes are equal, meaning the coefficients AA and BB in the equations of the lines will be the same. The only difference between the two equations is the constant term CC, which determines the vertical offset...
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Distance Formula from a Point to a Line in 3D In three-dimensional (3D) space , calculating the perpendicular distance from a point to a line is a common task in geometry, physics, and engineering. The distance formula for a point to a line in 3D space helps determine the shortest distance between a given point and a line, which is always perpendicular to the line. The Concept of Distance in 3D Consider a point P ( x 1 , y 1 , z 1 ) P(x_1, y_1, z_1) and a line defined by a point P 0 ( x 0 , y 0 , z 0 ) P_0(x_0, y_0, z_0) on the line and a direction vector v = ( v x , v y , v z ) \mathbf{v} = (v_x, v_y, v_z) that gives the direction of the line. The goal is to find the shortest distance between the point and the line. This shortest distance is the perpendicular distance from the point to the line. Equation of the Line A line in 3D space can be represented parametrically. If P 0 ( x 0 , y 0 , z 0 ) P_0(x_0, y_0, z_0) is a point on the line and v = ( v x , v y , v z ) \mathbf{v} = (v...
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