Engineering Calculus Notes 91

Engineering Calculus Notes 91 - 79 1.6. CROSS PRODUCTS...

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Unformatted text preview: 79 1.6. CROSS PRODUCTS Proposition 1.6.1. The 2 × 2 determinant x1 y 1 x2 y 2 is the signed area of the parallelogram OP RQ, where −→ − → → O P = x1 − + y 1 − ı −→ − → → O Q = x2 − + y 2 − ı and −→ −→ −→ − − − OR = OP + OQ. →→ Let us note several properties of the determinant ∆ (− , − ) which make it vw a useful computational tool. The proof of each of these properties is a straightforward calculation (Exercise 6): →→ Proposition 1.6.2. The 2 × 2 determinant ∆ (− , − ) has the following vw algebraic properties: →→→ vvw 1. It is additive in each slot:14 for any three vectors − 1 , − 2 , − ∈ R2 → →→ →→ →→ ∆ (− 1 + − , − 2 ) = ∆ (− 1 , − 2 ) + ∆ (− , − 2 ) v wv vv wv →→ − , − + − ) = ∆ (− , − ) + ∆ (− , − ) . → →→ →→ ∆( v v w vv vw 1 2 1 2 1 →→ 2. It is homogeneous in each slot: for any two vectors − 1 , − 2 ∈ R2 vv and any scalar r ∈ R →→ →→ →→ ∆ (r − 1 , − 2 ) = r ∆ (− 1 , − 2 ) = ∆ (− 1 , r − 2 ) . vv vv v v →→ 3. It is skew-symmetric: for any two vectors − 1 , − 2 ∈ R2 vv →→ →→ ∆ (− 2 , − 1 ) = −∆ (− 1 , − 2 ) . vv vv In particular, Corollary 1.6.3. A 2 × 2 determinant equals zero precisely if its rows are linearly dependent. 14 This is a kind of distributive law. ...
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