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Quaternions and Reflections

See how 4 dimensional quaternions involve vectors in 3-space and how the quaternion function F(v) = nvn gives a simple algebraic method of working with reflections in planes in 3-space.

Walls

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Let $P_1$ be a plane through the points $A(2,1,0), B(1,1,1)$ and $C(1,7,3)$ and $P_2$ be a plane through the points $A$, $B$ and $V(x,y,z)$. Find all the points $V(x,y,z)$ such that the two planes are perpendicular.