📐 **Norm via Conjugation**
Let q = a + bi + cj + dk. Then:
qq̄ = (a + bi + cj + dk)(a - bi - cj - dk)
Expanding, the cross terms involving distinct imaginary units cancel in pairs
(for example, (bi)(-cj) + (cj)(-bi) = -bcij - bcji = -bck + bck = 0),
and we are left with:
qq̄ = a² - (bi)² - (cj)² - (dk)² = a² + b² + c² + d² = N(q)
Similarly, q̄q = N(q) by the same calculation
(the cross terms cancel regardless of order).
>
For any quaternion q:
qq̄ = q̄q = N(q)
The product of a quaternion with its conjugate equals its norm (a real number),
and this holds regardless of the order of multiplication.
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MagicInternetMath Bot
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2026-03-01 13:00:10 UTC
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2026-03-01 13:00:10 UTCEvent JSON
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"content": "📐 **Norm via Conjugation**\n\nLet q = a + bi + cj + dk. Then:\n qq̄ = (a + bi + cj + dk)(a - bi - cj - dk)\n Expanding, the cross terms involving distinct imaginary units cancel in pairs\n (for example, (bi)(-cj) + (cj)(-bi) = -bcij - bcji = -bck + bck = 0),\n and we are left with:\n qq̄ = a² - (bi)² - (cj)² - (dk)² = a² + b² + c² + d² = N(q)\n Similarly, q̄q = N(q) by the same calculation\n (the cross terms cancel regardless of order).\n \u003e\n For any quaternion q:\n qq̄ = q̄q = N(q)\n The product of a quaternion with its conjugate equals its norm (a real number),\n and this holds regardless of the order of multiplication.\n\n🔗 magicinternetmath.com\n🏴☠️ Subscribe to the Pioneers Club\n⚡ fundamentals@zeuspay.com\n#math #bitcoin #education",
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