13.4.Ex70-75

# 13.4.Ex70-75 - 372 C H A P T E R 13 V E C T O R G E O M E T...

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372 CHAPTER 13 VECTOR GEOMETRY (ET CHAPTER 12) SOLUTION We form the following matrix: 243 01 7 15 3 ¯ ¯ ¯ ¯ ¯ ¯ 24 15 We now form the diagonals which slant from left to right and the diagonals which slant from right to left and assign corresponding sign to each diagonal: 2 0 1 −−− +++ 4 1 5 3 7 3 2 0 1 4 1 5 We add the products for the diagonals with a positive sign and subtract the products for the diagonals with a negative sign. This gives ¯ ¯ ¯ ¯ ¯ ¯ 7 ¯ ¯ ¯ ¯ ¯ ¯ = 2 · 1 · 3 + 4 · ( 7 ) · ( 1 ) + 3 · 0 · 5 3 · 1 · ( 1 ) 2 · ( 7 ) · 5 4 · 0 · 3 = 107 70. Prove that v × w = v × u if and only if u = w + λ v for some scalar . Assume that v ±= 0 . Transferring sides and using the distributive law and the property of parallel vectors, we obtain the follow- ing equivalent equalities: v × w = v × u 0 = v × u v × w 0 = v × ( u w ) This holds if and only if there exists a scalar such that u w = v u = w + v 71. Use Eq. (10) to prove the Cauchy–Schwarz inequality: | v · w |≤k v kk w k Show that equality holds if and only if w is a multiple of v or at least one of v and w is zero. Transferring sides in Eq. (10) we get ( v · w ) 2 =k v k 2 k w k 2 −k v × w k 2 (1) Since k v × w k 2 0, we have ( v · w ) 2 ≤k v k 2 k w k 2 Taking the square root of both sides gives | v · w v kk w k Equality | v · w |=k v kk w k holds if and only if ( v · w ) 2 v k 2 k w k 2 , that is by (1), if and only if k v × w k= 0, or v × w = 0 . This is equivalent to w = v for some scalar ,or v = 0 (Theorem 3 (c)).

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13.4.Ex70-75 - 372 C H A P T E R 13 V E C T O R G E O M E T...

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