quaternios

7/23/2019 Quaternios

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R2

a + bi

(a, b).

R3

R3

w = (α,β,γ )

R3

w = α + βi + γj

1 = (1, 0, 0)

i = (0, 1, 0)

j = (0, 0, 1)

ij

R3

{1, i , j}

R3

ij = a + bi + cj,

a,b,c ∈ R

i

i(ij) = i(a + bi + cj) = ia + i2b + i(cj) = ai − b + c(ij).

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i2 j = − j = ai − b + c(ij).

ij

a + bi + cj

− j = ai − b + c(a + bi + cj) = ai − b + ca + cbi + c2 j.

(ca − b) + (a + cb)i + (c2 + 1) j = 0.

{1, i , j}

c2 + 1 = 0

c ∈ R

i2 = j2 = k2 = ijk = −1

R4

Q

R

Q = {x + yi + zj + wk | x, y, z ,w ∈ R}

(a1 +

b1i + c1 j + d1k) + (a2 + b2i + c2 j + d2k) = (a + a1) + (b + b1)i + (c + c1) j + (d + d1)k.

1, i ,j ,k

1i = i, 1 j = j, 1k = k;

i2 = j2 = k2 = −1;

ij = − ji = k;

jk = −kj = i;

ki = −ik = j.

Q

Q

Q

α0 +

α1i + α2 j + α3k

α1 = α2 = α3 = 0

α = α0 + α1i + α2 j + α3k ∈ Q

α

α = α0 − α1i − α2 j − α3k.

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α = α0 + α1i + α2 j + α3k

α

α = αα = α2

0 + α2

1 + α2

2 + α2

3.

α = α0 + α1i + α2 j + α3k ∈ Q

α = 0

α = 0

β = α/ α

αβ = α α

α =

α

α = 1.

β = α−1

K

= 2

u, v = 0

u, v ∈ K

(u, v)K

K

(u, v)K = {x + yi + zj + wk | x, y, z ,w ∈ K}.

i2 = u, j2 = v, k2 = −uv;

ij = − ji = k;

jk = −kj = i;

ki = −ik = j ;

(u, v)K

(u, v)K

K

(u, v)K

K = C

(u, v)K

u

v

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