Engineering Calculus Notes 103

# Engineering Calculus Notes 103 - 91 1.6 CROSS PRODUCTS 5...

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1.6. CROSS PRODUCTS 91 5. Suppose that in ABC the vector from B to A is −→ v and that from B to C is −→ w . Use the vector formula for the distance from A to BC on p. 52 to prove that the area of the triangle is given by A ( ABC ) = 1 2 radicalBig ( −→ w · −→ w )( −→ v · −→ v ) ( −→ v · −→ w ) 2 . 6. Prove Proposition 1.6.2 . 7. Use Proposition 1.6.1 to prove Corollary 1.6.3 . ( Hint: If the rows are linearly dependent, what does this say about the parallelogram O PRQ ?) 8. Show that the cross product is: (a) skew-symmetric: −→ v × −→ w = −→ w × −→ v (b) additive in each slot: ( −→ v 1 + −→ v 2 ) × −→ w = ( −→ v 1 × −→ w ) + ( −→ v 2 × −→ w ) (use skew-symmetry to take care of the other slot: this is a kind of distributive law) (c)
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