Write the general formula for the product of two quaternions

Write the general formula for the product of two quaternions x_0 + x_1 j + x_2 j + x_3 k and y_0 + y_i j + y_2 j + y_3 k.

Solution

Let x = x₀+x_1i+x_2j+x₃k and

y = y₀+y_1i+y_2j+y_3k

_{xy =} (x₀+x_1i+x_2j+x₃k)( y₀+y_1i+y_2j+y_3k)

_{= (}x₀( y₀+y_1i+y_2j+y_3k)+x_1i( y₀+y_1i+y_2j+y_3k) + x_2j( y₀+y_1i+y_2j+y_3k)+x_3k( y₀+y_1i+y_2j+y_3k)

= x₀y₀+x₀y₁i+x₀y₂j+x₀y₃k + x₁iy₀ +x₁iy₁i +x₁iy₂j +x₁iy₃k +x₂jy₀+x₂jy₁i+x₂jy₃k+ x₃ky₀ +x₃ky₁i+x₃ky₂j+x₃ky₃k

Quaternions satisfies the following rules

i² = j² = k² = -1

ij = -ji = k

jk =-kj = i

ki = -ik =j

Using the above rules we get

xy = (x₀y₀-x₁y₁-x₂y₂-x₃y₃) + ( x₀y₁+x₁y₀+x₂y₃-x₃y₂)i + (x₀y₂-x₁y₃+x₂y₀+x₃y₁)j + (x₀y₃+x₁y₂-x₂y₁+x₃y₀)k