The four matrices Sx, Sy, S and I are defined by 0 -1 where i2-1. Show that S I and SS iS,, and obtain similar results by permutting x, y and z. Given that v is a vector with Cartesian components (vx, Dy, Dz), the matrix S(v) is defined as Prove that, for general non-zero vectors a and b, S(a)S(b) a bl+iS(a x b). Without further calculation, deduce that S(a) and S(b) commute if and only if a and b are parallel vectors.

Question
The four matrices Sx, Sy, S and I are defined by
0 -1
where i2-1. Show that S I and SS iS,, and obtain similar results
by permutting x, y and z. Given that v is a vector with Cartesian components
(vx, Dy, Dz), the matrix S(v) is defined as
Prove that, for general non-zero vectors a and b,
S(a)S(b) a bl+iS(a x b).
Without further calculation, deduce that S(a) and S(b) commute if and only if a
and b are parallel vectors.
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Transcribed Image Text

The four matrices Sx, Sy, S and I are defined by 0 -1 where i2-1. Show that S I and SS iS,, and obtain similar results by permutting x, y and z. Given that v is a vector with Cartesian components (vx, Dy, Dz), the matrix S(v) is defined as Prove that, for general non-zero vectors a and b, S(a)S(b) a bl+iS(a x b). Without further calculation, deduce that S(a) and S(b) commute if and only if a and b are parallel vectors.

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