Working with Lines in 3 Dimensions

Understanding Lines in 3-Dimensions

A line in 3-dimensional space is defined by combining a point and a direction.
The position vector of the point on the line provides a position in space, while the direction vector indicates the direction of the line.
A line can be defined using the vector equation of a line, given as r = a + tb, where ‘a’ is the position vector of a specific point on the line, ‘b’ is the direction vector, ‘t’ is a scalar quantity, and ‘r’ is any point on the line.
As ‘t’ varies, ‘r’ traces out the line in both directions. Positive values of ‘t’ trace out the direction of ‘b’, while negative values trace in the opposite direction.

Working with Lines in 3D

The equation of a line in three dimensions can be represented in a parametric form as x = x1 + at, y = y1 + bt, and z = z1 + ct, where (x1, y1, z1) are coordinates of a point on the line and a, b, c are direction ratios of the line.
Angle between two lines: It can be found using the direction vectors of the two lines. It is given by the formula cos θ = (a1a2 + b1b2 + c1c2) / √((a1²+b1²+c1²) * (a2²+b2²+c2²)), where a1, b1, c1 and a2, b2, c2 are the direction ratios of the first and second line respectively.
Intersection of lines: Two lines in 3D intersect if they have a common point. To find out if this is the case, equate the parametric equations of the two lines and solve for parameters.

Properties of Lines in 3D

Two lines are parallel if their direction vectors are proportional, i.e., each component of one direction vector is a constant times the corresponding component of the other direction vector.
Two lines are perpendicular if their direction vectors are orthogonal, i.e., the dot product of the direction vectors is zero.
A line in three-dimensional space has no bounds, it extends infinitely in both directions along its direction vector.

Real-World Applications

Lines in 3D are used in a wide range of fields including architecture, design, and engineering to represent linear structures.
They are also a critical part of computer graphics and game design, providing a way to render 3D environments and objects.

Note

Remember to understand and be able to apply all terms in bold, as they are fundamental in dealing with lines in 3 dimensions. Algorithms for operations involving lines, like checking intersection or angle between lines, are essential tools when working with 3D geometries.