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# notes001 (41) - v j 2 = u ± v& u ± v by property 4 = u...

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MATH 2203 °Quiz 1 (Version 3) Solution August 25, 2010 NAME____________________________ Recall the following properties of dot products: If u and v are any vectors and k is any scalar, then 1) u ° v = v ° u 2) ( k u ) ° v = u ° ( k v ) = k ( u ° v ) 3) u ° ( v + w ) = u ° v + u ° w 4) u ° u = j u j 2 5) u ° 0 = 0 . Use the above properties (whichever are needed) to prove that if u and v are any vectors, then j u ± v j 2 = j u j 2 ± 2 u ° v + j v j 2 . You must write the proof step°by°step with a little note at the end of each line of the proof indicating which of the above ±ve properties you are using in that step of the proof. Remember to write ² = ³where needed! Proof: j u ± v j 2
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Unformatted text preview: v j 2 = ( u ± v ) & ( u ± v ) by property 4 = ( u + ( ± v )) & ( u + ( ± v )) by de±nition of u ± v = ( u + ( ± v )) & u + ( u + ( ± v )) & ( ± v ) by property 3 = u & ( u + ( ± v )) + ( ± v ) & ( u + ( ± v )) by property 1 = u & u + u & ( ± v ) + ( ± v ) & u + ( ± v ) & ( ± v ) by property 3 = u & u ± u & v ± v & u + v & v by property 2 = u & u ± u & v ± u & v + v & v by property 1 = j u j 2 ± 2 u & v + j v j 2 by property 4. This completes the proof. 1...
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