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Any distance formula, as the name implies, calculates the distance between two points (the length of the line segment). The Pythagoras theorem allows us to get the distance formula between two points in a two-dimensional plane, which may also be extended to find the distance between two points in a three-dimensional plane. In coordinate geometry, there are several different forms of distance formulas.
What exactly is the Distance Formula?
In coordinate geometry, we have a list of distance formulas that may be used to find the distance between two points, the distance between two parallel lines, the distance between two parallel planes, and so on. All of the distance formulas are listed below, and we’ll go over each one in detail in the sections that follow.
Calculate the Distance Between Two Points Using the Distance Formula
The distance between two points in a two-dimensional plane and three-dimensional space will be demonstrated. The Pythagoras theorem is used to derive both distance formulas.
Distance Between Two Points in 2D
The Euclidean distance formula calculator is used to find the distance between two locations in a two-dimensional plane. Consider two locations in the 2D plane, A(x1,y1)(x1,y1) and B(x2,y2), to obtain the formula (x2,y2). Assume the distance between A and B is ‘d.’
Distance Formula to Calculate the Distance from a Point to a Line
In this section, we’ll look at the distance formulas for 2D and 3D distances from a point to a line. Both formulas don’t appear to be the same.
Distance From a Point to a Line in 2D
The length of the perpendicular line segment drawn from the point to the line is the distance formula for calculating the distance between two points. Consider a point P(x1,y1) on a line L in a two-dimensional plane with the equation ax + by + c =0. (x1,y1).
Distance From a Point To a Line in 3D
Consider a point P (x0,y0,z0),Q(x1,y1,z1) and a line (L) in 3D whose equation is x−x1/a=y−y1/b=z−z1/c. Then the distance (d) from the point P to L is,
d=|PQ¯×s¯|/|s¯|, where
P = (x0,y0,z0)………………..(x0,y0,z0) is the given point from which we are finding the distance to the line L
Q = (x1,y1,z1)………………….(x1,y1,z1) is a point on the line (which is from the equation of the line)
PQ=(x1−x0,y1−y0,z1−z0)
¯s. ……………………… = <a, b, c> is the direction vector of the line
PQׯs……………….. is the cross product of PQ¯ and ¯s.
Distance Formula to Calculate Distance Between Two Lines
In 2D, the distance between two parallel lines
The slopes of two parallel lines are usually the same, as we know. Assume L1: ax + by + c1 = 0 and L2: ax + by + c2 = 0 for the two parallel lines. The distance (d) between L1 and L2 is then as follows:
d = |c2−c1|/√a2+b2
Between two skew lines, what is the shortest distance?
If two lines in three-dimensional space are non-parallel and non-intersecting, they are said to be skew lines. Depending on whether they are given in cartesian or vector form, one of the following distance formulae can be used to compute the shortest distance between them.
Using the Distance Formula, find the distance between two parallel planes.
The formula for the distance between two lines is similar to the formula for the distance between two parallel lines. The normal vectors of two parallel planes are either equal or proportional, as we know. Thus, we can assume the equations of two parallel planes to be ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 to determine the distance formula between two parallel planes. Then (d) is the distance between two parallel surfaces.
d = |d2−d1|/√a2+b2+c2
Applications of Distance Formula
The distance formula has a wide range of applications in mathematics as well as in real-life situations. The following are some examples of how the distance formula can be used.
The distance formula can be used to compute the distance between any two points.
• The arg-and plane is used to represent a complex number, and the distance formula is used to calculate its magnitude.
You may use the distance formula to calculate the distance between two locations in three-dimensional and n-dimensional planes.
• The magnitude formula can be taken from the distance formula to determine the magnitude of a vector.
• The distance between two points in a sea can be calculated using the distance formula and the geographic coordinates of the two points.
• The shortest distance between two cities for air travel is calculated using the distance formula.
Conclusion
Now that you have all of the necessary information about the distance formula calculator and the applications of the distance formula, we hope you find this extensive information useful.
