Area and Volume study guide

Area and Volume study guide - B ′ by the geometric...

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Determinants 2. Area and Volume Area and volume interpretation of the determinant: (1) ±- a b 1 1 a b 2 2 = area of parallelogram with edges A = ( a 1 , a 2 ) , B = ( b 1 , b 2 ) . a 1 a 2 a 3 (2) ± b 1 b 2 b 3 = volume of parallelepiped with edges row-vectors A , B , C . c 1 c 2 c 3 In each case, choose the sign which makes the left side non-negative. Proof of (1). We begin with two preliminary observations. Let θ be the positive angle from A to B ; we assume it is < π , so that A and B have the general positions illustrated. Let θ = π / 2 θ , as illustrated. Then cos θ = sin θ . Draw the vector B obtained by rotating B to the right by π / 2. The picture shows that B = ( b 2 , b 1 ), and | B | = | B | . To prove (1) now, we have a standard formula of Euclidean geometry, area of parallelogram = | A || B | sin θ = | A || B | cos θ , by the above observations 2 = A ·- B ,
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Unformatted text preview: B ′ , by the geometric deFnition of dot product = a 1 b 2 − a 2 b 1 by the formula for B ′ This proves the area interpretation (1) if A and B have the position shown. If their positions are reversed, then the area is the same, but the sign of the determinant is changed, so the formula has to read, ± a 1 a 2 ± area of parallelogram = ±-, whichever sign makes the right side ≥-. ± b 1 b 2 ± The proof of the analogous volume formula (2) will be made when we study the scalar triple product A ·-B ×-C . Generalizing (1) and (2), n ×-n determinants can be interpreted as the hypervolume in n-space of a n-dimensional parallelotope. θ A B A B C θ θ θ θ A B A B b b b 1 2 1 b B B 1...
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