# Given u 372 v 621 and w 238 verify the following

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12. Given u =(− 3,7,2 ) , v = ( 6,2,1 ) , and w =(− 2,3,8 ) , verify the following arithmetic properties of vectors. (12 marks: 4 marks each) a) ( v + w ) = v + Step #1: Calculate the left side of the equation – vector addition first. w
( v + w ) = ( 3,7,2 ) × [ ( 6,2,1 ) + ( 2,3,8 ) ] ( v + w ) = ( 3,7,2 ) × [ ( ( 6 + ( 2 ) ) , ( 2 + 3 ) , ( 1 + 8 ) ) ] ( v + w ) = ( 3,7,2 ) × ( 4,5,9 ) Step #2: Calculate the left side of the equation – calculate the cross product. ¿ Let ( v + w ) = p =( 4,5,9 ) p = u 2 p 3 p 2 u 3 ,u 3 p 1 p 3 u 1 ,u 1 p 2 p 1 u 2 p =( ( 7 ) ( 9 ) ( 5 ) ( 2 ) , ( 9 ) ( 4 ) ( 9 ) ( 3 ) , ( 3 ) ( 5 ) ( 4 ) ( 7 ) ) p = ( ( 63 10 ) , ( 8 −(− 27 ) ) , ( 15 28 ) ) p = ( 53,35, 43 ) ( v + w ) = ( 53,35, 43 ) Therefore, the cross product is ( v + w ) = ( 53,35, 43 ) . Step #3: Calculate the right side of the equation – calculate cross product of v . v = ( 3,7,2 ) × ( 6,2,1 ) v = u 2 v 3 v 2 u 3 ,u 3 v 1 v 3 u 1 ,u 1 v 2 v 1 u 2 v =( ( 7 ) ( 1 ) ( 2 ) ( 2 ) , ( 2 ) ( 6 ) ( 1 ) ( 3 ) , ( 3 ) ( 2 ) ( 6 ) ( 7 ) ) v = ( ( 7 4 ) , ( 12 −(− 3 ) ) , ( 6 42 ) ) v =( 3,15, 48 )
Step #4: Calculate the right side of the equation – calculate cross product of v .
b) u∙ ( v + w ) = u∙ v + u∙ Step #1: Calculate the left side of the equation – vector addition first. w
Step #2: Calculate the left side of the equation – calculate dot product of u∙ ( v + w ) . . . .
u∙ v + u∙ w = 41 Therefore, LS = RS and u∙ ( v + w ) = u∙ v + u∙ w . c) ( u + v ) ( u + v ) = | u | 2 + | v | 2 + 2 ( u∙ v ) Step #1: Calculate the left side of the equation – vector addition of u + v . ( u + v ) = [ ( 3,7,2 ) + ( 6,2,1 ) ] 3 + 6 ¿ 2 + 1 ¿ ( u + v ) = ¿ ( u + v ) =( 3,9,3 ) Step #2: Calculate the left side of the equation – vector addition of u + v . ( u + v ) ( u + v ) = ( 3,9,3 ) ( 3,9,3 ) ( u + v ) ( u + v ) =[ ( 3 ) ( 3 ) + ( 9 ) ( 9 ) + ( 3 ) ( 3 ) ] ( u + v ) ( u + v ) = 9 + 81 + 9 ( u + v ) ( u + v ) = 99 Step #3: Calculate the right side of the equation – calculate the dot product of 2 ( u∙ v ) . 2 ( u∙ v ) = 2 ( 3,7,2 ) ( 6,2,1 )
2 ( u∙ v ) = 2 ( ( 3 ) ( 6 ) + ( 7 ) ( 2 ) +( 2 )( 1 ) ) 2 ( u∙ v ) = 2 (− 18 + 14 + 2 ) 2 ( u∙ v ) = 2 (− 2 ) 2 ( u∙ v ) =− 4 Step #4: Calculate the right side of the equation – vector addition of | u | 2 + | v | 2 . | u | 2 + | v | 2 = ( ( 3 ) 2 + ( 7 ) 2 + ( 2 ) 2 ) 2 + ( ( 6 ) 2 + ( 2 ) 2 + ( 1 ) 2 ) 2 | u | 2 + | v | 2 = ( 9 + 49 + 4 ) 2 + ( 36 + 4 + 1 ) 2 | u | 2 + | v | 2 = 62 + 41 | u | 2 + | v | 2 = 103