<Dd> − ∑ i = 1 n m i (Δ r i × (Δ r i × u)) = − ∑ i = 1 n m i ((0 − Δ r 3, i Δ r 2, i Δ r 3, i 0 − Δ r 1, i − Δ r 2, i Δ r 1, i 0) ((0 − Δ r 3, i Δ r 2, i Δ r 3, i 0 − Δ r 1, i − Δ r 2, i Δ r 1, i 0) (u 1 u 2 u 3)))... cross-product as matrix multiplication = − ∑ i = 1 n m i ((0 − Δ r 3, i Δ r 2, i Δ r 3, i 0 − Δ r 1, i − Δ r 2, i Δ r 1, i 0) (− Δ r 3, i u 2 + Δ r 2, i u 3 + Δ r 3, i u 1 − Δ r 1, i u 3 − Δ r 2, i u 1 + Δ r 1, i u 2)) = − ∑ i = 1 n m i (− Δ r 3, i (+ Δ r 3, i u 1 − Δ r 1, i u 3) + Δ r 2, i (− Δ r 2, i u 1 + Δ r 1, i u 2) + Δ r 3, i (− Δ r 3, i u 2 + Δ r 2, i u 3) − Δ r 1, i (− Δ r 2, i u 1 + Δ r 1, i u 2) − Δ r 2, i (− Δ r 3, i u 2 + Δ r 2, i u 3) + Δ r 1, i (+ Δ r 3, i u 1 − Δ r 1, i u 3)) = − ∑ i = 1 n m i (− Δ r 3, i 2 u 1 + Δ r 1, i Δ r 3, i u 3 − Δ r 2, i 2 u 1 + Δ r 1, i Δ r 2, i u 2 − Δ r 3, i 2 u 2 + Δ r 2, i Δ r 3, i u 3 + Δ r 2, i Δ r 1, i u 1 − Δ r 1, i 2 u 2 + Δ r 3, i Δ r 2, i u 2 − Δ r 2, i 2 u 3 + Δ r 3, i Δ r 1, i u 1 − Δ r 1, i 2 u 3) = − ∑ i = 1 n m i (− (Δ r 2, i 2 + Δ r 3, i 2) u 1 + Δ r 1, i Δ r 2, i u 2 + Δ r 1, i Δ r 3, i u 3 + Δ r 2, i Δ r 1, i u 1 − (Δ r 1, i 2 + Δ r 3, i 2) u 2 + Δ r 2, i Δ r 3, i u 3 + Δ r 3, i Δ r 1, i u 1 + Δ r 3, i Δ r 2, i u 2 − (Δ r 1, i 2 + Δ r 2, i 2) u 3) = − ∑ i = 1 n m i (− (Δ r 2, i 2 + Δ r 3, i 2) Δ r 1, i Δ r 2, i Δ r 1, i Δ r 3, i Δ r 2, i Δ r 1, i − (Δ r 1, i 2 + Δ r 3, i 2) Δ r 2, i Δ r 3, i Δ r 3, i Δ r 1, i Δ r 3, i Δ r 2, i − (Δ r 1, i 2 + Δ r 2, i 2)) (u 1 u 2 u 3) = − ∑ i = 1 n m i (Δ r i) 2 u − ∑ i = 1 n m i (Δ r i × (Δ r i × u)) = (− ∑ i = 1 n m i (Δ r i) 2) u...u is not characteristic of P i (\ displaystyle (\ begin (aligned) - \ sum _ (i = 1) ^ (n) m_ (i) ((\ boldsymbol (\ Delta)) \ mathbf (r) _ (i) \ times ((\ boldsymbol (\ Delta)) \ mathbf (r) _ (i) \ times \ mathbf (u))) & = - \ sum _ (i = 1) ^ (n) m_ (i) \ left ((\ begin (bmatrix) 0& - \ Delta r_ (3, i) & \ Delta r_ (2, i) \ \ \ Delta r_ (3, i) &0& - \ Delta r_ (1, i) \ \ - \ Delta r_ (2, i) & \ Delta r_ (1, i) &0 \ end (bmatrix)) \ left ((\ begin (bmatrix) 0& - \ Delta r_ (3, i) & \ Delta r_ (2, i) \ \ \ Delta r_ (3, i) &0& - \ Delta r_ (1, i) \ \ - \ Delta r_ (2, i) & \ Delta r_ (1, i) &0 \ end (bmatrix)) (\ begin (bmatrix) u_ (1) \ \ u_ (2) \ \ u_ (3) \ end (bmatrix)) \ right) \ right) \; \ ldots (\ text (cross-product as matrix multiplication)) \ \ (6pt) & = - \ sum _ (i = 1) ^ (n) m_ (i) \ left ((\ begin (bmatrix) 0& - \ Delta r_ (3, i) & \ Delta r_ (2, i) \ \ \ Delta r_ (3, i) &0& - \ Delta r_ (1, i) \ \ - \ Delta r_ (2, i) & \ Delta r_ (1, i) &0 \ end (bmatrix)) (\ begin (bmatrix) - \ Delta r_ (3, i) \, u_ (2) + \ Delta r_ (2, i) \, u_ (3) \ \ + \ Delta r_ (3, i) \, u_ (1) - \ Delta r_ (1, i) \, u_ (3) \ \ - \ Delta r_ (2, i) \, u_ (1) + \ Delta r_ (1, i) \, u_ (2) \ end (bmatrix)) \ right) \ \ (6pt) & = - \ sum _ (i = 1) ^ (n) m_ (i) (\ begin (bmatrix) - \ Delta r_ (3, i) (+ \ Delta r_ (3, i) \, u_ (1) - \ Delta r_ (1, i) \, u_ (3)) + \ Delta r_ (2, i) (- \ Delta r_ (2, i) \, u_ (1) + \ Delta r_ (1, i) \, u_ (2)) \ \ + \ Delta r_ (3, i) (- \ Delta r_ (3, i) \, u_ (2) + \ Delta r_ (2, i) \, u_ (3)) - \ Delta r_ (1, i) (- \ Delta r_ (2, i) \, u_ (1) + \ Delta r_ (1, i) \, u_ (2)) \ \ - \ Delta r_ (2, i) (- \ Delta r_ (3, i) \, u_ (2) + \ Delta r_ (2, i) \, u_ (3)) + \ Delta r_ (1, i) (+ \ Delta r_ (3, i) \, u_ (1) - \ Delta r_ (1, i) \, u_ (3)) \ end (bmatrix)) \ \ (6pt) & = - \ sum _ (i = 1) ^ (n) m_ (i) (\ begin (bmatrix) - \ Delta r_ (3, i) ^ (2) \, u_ (1) + \ Delta r_ (1, i) \ Delta r_ (3, i) \, u_ (3) - \ Delta r_ (2, i) ^ (2) \, u_ (1) + \ Delta r_ (1, i) \ Delta r_ (2, i) \, u_ (2) \ \ - \ Delta r_ (3, i) ^ (2) \, u_ (2) + \ Delta r_ (2, i) \ Delta r_ (3, i) \, u_ (3) + \ Delta r_ (2, i) \ Delta r_ (1, i) \, u_ (1) - \ Delta r_ (1, i) ^ (2) \, u_ (2) \ \ + \ Delta r_ (3, i) \ Delta r_ (2, i) \, u_ (2) - \ Delta r_ (2, i) ^ (2) \, u_ (3) + \ Delta r_ (3, i) \ Delta r_ (1, i) \, u_ (1) - \ Delta r_ (1, i) ^ (2) \, u_ (3) \ end (bmatrix)) \ \ (6pt) & = - \ sum _ (i = 1) ^ (n) m_ (i) (\ begin (bmatrix) - (\ Delta r_ (2, i) ^ (2) + \ Delta r_ (3, i) ^ (2)) \, u_ (1) + \ Delta r_ (1, i) \ Delta r_ (2, i) \, u_ (2) + \ Delta r_ (1, i) \ Delta r_ (3, i) \, u_ (3) \ \ + \ Delta r_ (2, i) \ Delta r_ (1, i) \, u_ (1) - (\ Delta r_ (1, i) ^ (2) + \ Delta r_ (3, i) ^ (2)) \, u_ (2) + \ Delta r_ (2, i) \ Delta r_ (3, i) \, u_ (3) \ \ + \ Delta r_ (3, i) \ Delta r_ (1, i) \, u_ (1) + \ Delta r_ (3, i) \ Delta r_ (2, i) \, u_ (2) - (\ Delta r_ (1, i) ^ (2) + \ Delta r_ (2, i) ^ (2)) \, u_ (3) \ end (bmatrix)) \ \ (6pt) & = - \ sum _ (i = 1) ^ (n) m_ (i) (\ begin (bmatrix) - (\ Delta r_ (2, i) ^ (2) + \ Delta r_ (3, i) ^ (2)) & \ Delta r_ (1, i) \ Delta r_ (2, i) & \ Delta r_ (1, i) \ Delta r_ (3, i) \ \ \ Delta r_ (2, i) \ Delta r_ (1, i) & - (\ Delta r_ (1, i) ^ (2) + \ Delta r_ (3, i) ^ (2)) & \ Delta r_ (2, i) \ Delta r_ (3, i) \ \ \ Delta r_ (3, i) \ Delta r_ (1, i) & \ Delta r_ (3, i) \ Delta r_ (2, i) & - (\ Delta r_ (1, i) ^ (2) + \ Delta r_ (2, i) ^ (2)) \ end (bmatrix)) (\ begin (bmatrix) u_ (1) \ \ u_ (2) \ \ u_ (3) \ end (bmatrix)) \ \ & = - \ sum _ (i = 1) ^ (n) m_ (i) (\ Delta r_ (i)) ^ (2) \ mathbf (u) \ \ (6pt) - \ sum _ (i = 1) ^ (n) m_ (i) ((\ boldsymbol (\ Delta)) \ mathbf (r) _ (i) \ times ((\ boldsymbol (\ Delta)) \ mathbf (r) _ (i) \ times \ mathbf (u))) & = \ left (- \ sum _ (i = 1) ^ (n) m_ (i) (\ Delta r_ (i)) ^ (2) \ right) \ mathbf (u) \; \ ldots \; \ mathbf (u) (\ text (is not characteristic of)) P_ (i) \ end (aligned))) </Dd> <P> Finally, the result is used to complete the main proof as follows: </P> <Dl> <Dd> τ = − ∑ i = 1 n m i (Δ r i × (Δ r i × α)) + ω × − ∑ i = 1 n m i Δ r i × (Δ r i × ω)) = (− ∑ i = 1 n m i (Δ r i) 2) α + ω × (− ∑ i = 1 n m i (Δ r i) 2) ω (\ displaystyle (\ begin (aligned) (\ boldsymbol (\ tau)) & = - \ sum _ (i = 1) ^ (n) m_ (i) ((\ boldsymbol (\ Delta)) \ mathbf (r) _ (i) \ times ((\ boldsymbol (\ Delta)) \ mathbf (r) _ (i) \ times (\ boldsymbol (\ alpha)))) + (\ boldsymbol (\ omega)) \ times - \ sum _ (i = 1) ^ (n) m_ (i) (\ boldsymbol (\ Delta)) \ mathbf (r) _ (i) \ times ((\ boldsymbol (\ Delta)) \ mathbf (r) _ (i) \ times (\ boldsymbol (\ omega)))) \ \ & = \ left (- \ sum _ (i = 1) ^ (n) m_ (i) (\ Delta r_ (i)) ^ (2) \ right) (\ boldsymbol (\ alpha)) + (\ boldsymbol (\ omega)) \ times \ left (- \ sum _ (i = 1) ^ (n) m_ (i) (\ Delta r_ (i)) ^ (2) \ right) (\ boldsymbol (\ omega)) \ end (aligned))) </Dd> </Dl> <Dd> τ = − ∑ i = 1 n m i (Δ r i × (Δ r i × α)) + ω × − ∑ i = 1 n m i Δ r i × (Δ r i × ω)) = (− ∑ i = 1 n m i (Δ r i) 2) α + ω × (− ∑ i = 1 n m i (Δ r i) 2) ω (\ displaystyle (\ begin (aligned) (\ boldsymbol (\ tau)) & = - \ sum _ (i = 1) ^ (n) m_ (i) ((\ boldsymbol (\ Delta)) \ mathbf (r) _ (i) \ times ((\ boldsymbol (\ Delta)) \ mathbf (r) _ (i) \ times (\ boldsymbol (\ alpha)))) + (\ boldsymbol (\ omega)) \ times - \ sum _ (i = 1) ^ (n) m_ (i) (\ boldsymbol (\ Delta)) \ mathbf (r) _ (i) \ times ((\ boldsymbol (\ Delta)) \ mathbf (r) _ (i) \ times (\ boldsymbol (\ omega)))) \ \ & = \ left (- \ sum _ (i = 1) ^ (n) m_ (i) (\ Delta r_ (i)) ^ (2) \ right) (\ boldsymbol (\ alpha)) + (\ boldsymbol (\ omega)) \ times \ left (- \ sum _ (i = 1) ^ (n) m_ (i) (\ Delta r_ (i)) ^ (2) \ right) (\ boldsymbol (\ omega)) \ end (aligned))) </Dd>

Moment of inertia is the rotational equivalent of mass