Derive and verify vector identities

TensorExpand[(v⨯w).v == 0]

Cross[v, w] . v == 0

TensorExpand[a⨯(b⨯c) == b(a.c) - c (a.b)]

Cross[a, Cross[b, c]] == -(c*a . b) + b*a . c

TensorExpand[(a⨯b).(c⨯d) == a.c b.d - a.d b.c]

Cross[a, b] . Cross[c, d] == -(a . d*b . c) + a . c*b . d

TensorExpand[a (b⨯c).d - b(a⨯c).d + c(a⨯b).d - d(a⨯b).c == 0]

-(d*Cross[a, b] . c) + c*Cross[a, b] . d - b*Cross[a, c] . d + a*Cross[b, c] . d == 0

TensorExpand[(a⨯b)⨯(c⨯d)]

Cross[Cross[a, b], Cross[c, d]]

TensorExpand[a⨯b⨯c == b⨯c⨯a]

Cross[a, b, c] == Cross[b, c, a]

TensorExpand[(a⨯b⨯c).(d⨯e⨯f)]

Cross[a, b, c] . Cross[d, e, f]

$Line = 0

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