The 'recipe' is given in the question. You need to know that the scalar product of two perpendicular vectors is 0 so that, if one vector lies in a plane and the other is normal to the plane, then their scalar product is zero. This gives the equation of a plane through the origin in

R^{3}

v \cdot n = a x + b y + c z = 0

. The diagram should help you to visualise that, if

u_{0}

is on the plane and

n

is a vector normal to the plane, then the points

u_{0} + t n

and

u_{0} - t n

are reflections of each other in the plane.

Where quaternions are equivalent to vectors we are not using boldface fonts other than in introducing the unit vectors $i, j, k$ along the axes in $R^{3}$ .

The quaternion functions and quaternion algebra give a neat and efficient way to work with reflections in $R^{3}$ and they are very useful in computer graphics programs.