Submit u=(u1,u2,u3), v=(v1,v2,v3).
Calculate vector product of u and v:
u×v=∣∣iu1v1ju2v2ku3v3∣∣=i(u2v3−u3v2)+j(u1v3−u3v1)+k(u1v2−u2v1)==(u2v3−u3v2,u1v3−u3v1,u1v2−u2v1)..
And obtain:
u×v×u=∣∣iu2v3−u3v2u1ju1v3−u3v1u2ku1v2−u2v1u3∣∣=i(u1u3v3−u32v1−−u1u2v2+u22v1)+j(u2u3v3−u32v2−u12v2+u1u2v1)+k(u22v3−u2u3v2−−u12v3+u1u3v1)=(u1u3v3−u32v1−u1u2v2+u22v1,u2u3v3−u32v2−−u12v2+u1u2v1,u22v3−u2u3v2−u12v3+u1u3v1).
According to vector product properties u×v×u=u×(v×u)=−(v×u)×u, so
v×u×u=−u×v×u=−(u1u3v3−u32v1−u1u2v2+u22v1,u2u3v3−u32v2−−u12v2+u1u2v1,u22v3−u2u3v2−u12v3+u1u3v1)=(−u1u3v3+u32v1+u1u2v2−−u22v1,−u2u3v3+u32v2+u12v2−u1u2v1,−u22v3+u2u3v2+u12v3−u1u3v1).
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