- 2075 Baisakh · 5 marks
Scalar triple product identity
Short answer
Use (c×d)×(e×f) = [c e f] d − [d e f] c. Dot with (a×b) gives [a×b c×d e×f] = [a b d][c e f] − [a b c][d e f].
Prove that [a×b c×d e×f] = [a b d][c e f] − [a b c][d e f].
We use [u v w] = (u × v) · w and (P × Q) × R = Q(P · R) − P(Q · R).
Step 1: Expand (c × d) × (e × f)
Put Y = e × f. Then
(c × d) × Y = d(c · Y) − c(d · Y), with c · Y = c · (e × f) = [c e f] and d · Y = [d e f].
So (c × d) × (e × f) = [c e f] d − [d e f] c.
Step 2: Take the triple product
[a×b c×d e×f] = (a × b) · [(c × d) × (e × f)]
= (a × b) · ([c e f] d − [d e f] c)
= [c e f] (a × b) · d − [d e f] (a × b) · c
= [c e f] [a b d] − [d e f] [a b c]
Hence [a×b c×d e×f] = [a b d][c e f] − [a b c][d e f].
(Check with a = i, b = j, c = i, d = k, e = j, f = k: a×b = k, c×d = −j, e×f = i, so the left side is k · ((−j) × i) = k · k = 1. The right side is [i j k][i j k] − [i j i][k j k] = 1 − 0 = 1 ✓.)
✓ Answer checked by Claude Opus 5.5 (high effort)
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