- A-Level Further Maths

# Vectors

The equation of a line can be represented by vectors in 3D. The line that passes through **A** and **R** can be written as:

r=a+ λb

where

__a__is the position vector of a known point on the line__b__is the**direction vector**of the line (a vector parallel to the line)__r__is the position vector of any arbitrary point on the lineλ is a scalar parameter

If you need to find the equation of a line from just two points, **C** and **D**, and you do not know the direction vector, use the fact that the vector between the two points (__d__ - __c__) is the direction vector of the line:

The equation of this line is given as

r=c+ λ(d-c)

### Equations of 3D line in Cartesian Form

If you need to find the equation of a straight three dimensional line in Cartesian form (in terms of x, y and x), convert using the vectors __a__ and __b__ from the vector equation __r__ = __a__ + λ__b.__