Shortest distance between two parallel lines in 3 dimensional geometry | NCERT Maths Class 12th

Derivation of the formula to find the Shortest distance between two parallel lines in 3 dimensional geometry.

No need to memorize when you an understand the derivation.

Important 3D concepts -

The cross product of two vectors a and b is defined only in three-dimensional space and is denoted by a × b. If the vectors a and b are parallel (i.e., the angle θ between them is either 0° or 180°), by the formula you will see in the video, the cross product of a and b is the zero vector 0.

The dot product of two vectors is the product of the magnitude of the two vectors and the cos of the angle between them.

Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point). This is the informal meaning of the term dimension.

In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space. When n = 3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3. This serves as a three-parameter model of the physical universe (that is, the spatial part, without considering time) in which all known matter exists. However, this space is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the three values refer to measurements in different directions (coordinates), any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space (plane). Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and length.


Direction cosine
In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three coordinate axes. Equivalently, they are the contributions of each component of the basis to a unit vector in that direction. Direction cosines are an analogous extension of the usual notion of slope to higher dimensions.


Direction Ratios
If a, b, c are three numbers proportional to the direction cosine l, m, n of a straight line, then a, b, c are called its direction ratios. They are also called direction numbers or direction components.



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