What is the vector equation of a line, L, in 3-D? Write this both as a vector with vector labels, r, and also in vector components.

Answer 1

The vector equation of a line L in 3-D can be written as:

r = a + λd

where:
- r is the position vector of any point on the line
- a is the position vector of a known point on the line
- d is the direction vector of the line
- λ is a scalar parameter that can take any real value

In vector components, the equation becomes:

r = <x, y, z> = <x0, y0, z0> + λ<d1, d2, d3>

where:
- <x0, y0, z0> is the known point on the line
- <d1, d2, d3> is the direction vector of the line.