#### You may also like A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?  To add and take scalar multiples of quaternions just treat them like 4-dimensional vectors, for example: $$(a_1 +b_1{\bf i} + c_1{\bf j} + d_1{\bf k}) + (a_2 +b_2{\bf i} + c_2{\bf j} + d_2{\bf k})= (a_1+a_2) + (b_1+b_2){\bf i} + (c_1+c_2){\bf j} + (d_1+d_2){\bf k}).$$ Multiplication is defined by the rules of ordinary algebra where $${\bf i^2}={\bf j^2}={\bf k^2} =-1,\quad {\bf i j} = {\bf k} = {\bf -j i}, \quad {\bf j k} = {\bf i} = {\bf -k j},\quad {\rm and}\quad {\bf k i}= {\bf j} = {\bf -i k}.$$ For example $$(2 + 3{\bf i} +4{\bf j} +5{\bf k})(6 + 7{\bf i} + 8{\bf j} + 9{\bf k}) = (12 - 21 - 32 - 45) + (36-40){\bf i} + (35-27){\bf j} + (21 - 28){\bf k} = -86 - 4{\bf i} +8{\bf j} -7 {\bf k}.$$ To read about number systems, where quaternions fit in, why there are no three-dimensional numbers and numbers in higher dimensions, see the NRICH article What Are Numbers?