Ch_03 - Vector I - Exam_Soln

Ch_03 - Vector I - Exam_Soln - Exam Problems of Vector...

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Exam Problems of Vector Differential Calculus
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Vector Differential Calculus Fundamentals Fundamentals 1. (05’) Prove (A) () ( ) ( )   ABC B A C C A B  (B) ( ) ( ) ( ) ( )  AB CD ACBD ADBC (C) ( ) ( ) (   A B C D ACDB BCDA ABDC ABCD Solution: (A) ( ) ( ) A A B [ ( ) ( ) ] ( (        ABC BCA B A A B BAC CAB (B) ( ) ( ) ( ) ( ) Let  XA B , then XCD { } { ( ( )} ( )( ) ( )( ) AB C D D AC BD AD BC (C) ( By middle factor rule,  ( ) ( ) ( ( )) ( ( )) ACDB BCDA or   ( ) ( ) (( ) ) (( ) ) ABDC ABCD
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Vector Differential Calculus Fundamentals (Continued) (or for (A) () ( ) ( )   ABC B A C C A B  Set  123 aaa Aijk ,  bbb Bi jk , ccc Cijk  1 2 3 23 13 12 32 31 21 LHS aa a bbb bb bb bb aa a cc cc cc bc      ij k ijk i j k k 212 221 313 331 112 121 323 332 113 131 223 232 LHS abc abc abc a b ca b b b c   i j k 11 22 33 ac AC ab ab ab AB   2 2 1 2 3 1 2 3 RHS b b b c c c    ACB ABC RHS a b b b b c i j k We get LHS RHS
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Vector Differential Calculus Fundamentals 2. (06’) Prove (A) from  CA B , prove the law of cosine 222 2c o s B A B  where || A A , B B , and is angel between A and B . (B) Using the vectors cos sin Pi j , cos sin Qi j , cos sin Ri j Prove the familiar trigonometric identities. sin( ) sin cos cos sin cos( ) cos cos sin sin   Solution: (A) 2 22 () ( ) ( ) 2( ) | | | 2 | || | cos 2 cos C AB A B   CC ABAB AA BB A B  (B) On the polar coordinates 11 cos , sin cos , sin xy | | | | c o s ( ) (cos sin ) (cos sin ) sin sin    PQ P Q ijij Since c o s s i n 1  P , and c o s s i  Q cos sin 0 (sin cos cos sin ) cos sin 0  ij k k 2 | | || | sin( ) cos sin ) cos sin
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Vector Differential Calculus Fundamentals 3. Given the vectors ' bc a ab c , ca b and ab c Show that if 0  , (A) ' 1  aa bb cc  (B) ' 0 ab ac  , 0 ba bc , 0 ca cb Solution:
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Vector Differential Calculus Fundamentals 4. Prove that the diagonals of a parallelogram bisect each other. Solution:
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Vector Differential Calculus Fundamentals 5. Find the equation of a straight line which passes through two given point A and B having position vectors a and b with respect to and origin O . Solution:
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Vector Differential Calculus Fundamentals 6. Consider a tetrahedron with faces 1234 , , , FFFF . Let , VVVV be vectors whose magnitudes are respectively equal to the area of and whose directions are perpendicular to these faces in the outward direction. Show that +++ 0 . Solution: The area of a triangular face determined by R and S is 1 2 || RS .
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This note was uploaded on 04/27/2011 for the course CHEM 101 taught by Professor Suh during the Spring '11 term at University of Toronto- Toronto.

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Ch_03 - Vector I - Exam_Soln - Exam Problems of Vector...

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