Product of Sums of Four Squares/Proof 2
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Theorem
Let $a, b, c, d, w, x, y, z$ be numbers.
Then:
\(\ds \) | \(\) | \(\ds \left({a^2 + b^2 + c^2 + d^2}\right) \left({w^2 + x^2 + y^2 + z^2}\right)\) | ||||||||||||
\(\ds =\) | \(\) | \(\ds \left({a w + b x + c y + d z}\right)^2\) | ||||||||||||
\(\ds \) | \(+\) | \(\ds \left({a x - b w + c z - d y}\right)^2\) | ||||||||||||
\(\ds \) | \(+\) | \(\ds \left({a y - b z - c w + d x}\right)^2\) | ||||||||||||
\(\ds \) | \(+\) | \(\ds \left({a z + b y - c x - d w}\right)^2\) |
Proof
Let:
- $\mathbf m = a \mathbf 1 + b \mathbf i + c \mathbf j + d \mathbf k$
- $\mathbf n = -w \mathbf 1 + x \mathbf i + y \mathbf j + z \mathbf k$
be two quaternions.
Then:
\(\ds \size {\mathbf m} \size {\mathbf n}\) | \(=\) | \(\ds \size {\mathbf m \mathbf n}\) | Quaternion Modulus of Product of Quaternions | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {a^2 + b^2 + c^2 + d^2} \paren {\paren {-w}^2 + x^2 + y^2 + z^2}\) | \(=\) | \(\ds \size {\mathbf m \mathbf n}\) | Definition of Quaternion Modulus | ||||||||||
\(\ds \) | \(=\) | \(\ds \paren {-a w - b x - c y - d z}^2\) | Definition of Quaternion Multiplication | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {a x - b w + c z - d y}^2\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {a y - b z - c w + d x}^2\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {a z + b y - c x - d w}^2\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {a^2 + b^2 + c^2 + d^2} \paren {w^2 + x^2 + y^2 + z^2}\) | \(=\) | \(\ds \paren {a w + b x + c y + d z}^2\) | simplifying | ||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {a x - b w + c z - d y}^2\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {a y - b z - c w + d x}^2\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \paren {a z + b y - c x - d w}^2\) |
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem