mathtools - Math Tools / Math Review 1 Resultant Vector...

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1 Math Tools / Math Review
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2 Resultant Vector Problem: We wish to find the vector sum of vectors A and B Pictorially, this is shown in the figure on the right. Mathematically, we want to break the vector into x, y, and maybe z components and find the resultant vector ) 12 6 ( ) 6 7 ( tan ) 6 7 ( ) 12 6 ( ˆ 6 ˆ 12 And ˆ 7 ˆ 6 Let 2 2 + - = = - + + = + - + = + = x y B A y x B y x A θ A B A + B θ
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3 System of equations Let 6x+7y=15 And 4x-3y=9 Now find x and y which satisfy these equations I use “method of minors” 46 28 18 ) 4 )( 7 ( ) 3 )( 6 ( 3 4 7 6 e Determinat - = - - = - - = - =
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4 System of Equations cont’d 347 . 2 46 ) 9 )( 7 ( ) 3 )( 15 ( det 3 9 7 15 = - - - = - = x The solution for x is found by creating a “minor” wherein the constants in the equation are substituted in place of the “x” value and the value of the minor is found. It is then divided by the value of the determinant. I then “back-substitute” the value of x into my initial equation and solve for y. 6(2.347)+7y=15 and solve for y (y=.1304)
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5 Or I could use the minors again 1304 . 0 46 ) 15 )( 4 ( ) 9 )( 6 ( det 9 4 15 6 = - - = = y
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6 Larger Systems det 10 12 15 2 15 60 10 7 12 3434 12 5 15 9 10 10 5 2 9 7 10 12 2 15 6 10 12 5 2 15 9 10 7 6 det - = - = - + - - - - = - - = x Larger systems are broken down into their resultant minors. For example: 6x + 7y +10z =12 -9x+15y+2z=60 5x +12y-10z=15 + - + - + - + - +
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7 Spherical Coordinates Cartesian coordinates: x, y, z Spherical coordinates: r, θ , φ Math Majors NOTE Theta! ( 29
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This note was uploaded on 02/12/2012 for the course PHYSICS 104 taught by Professor Staff during the Fall '10 term at Rutgers.

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mathtools - Math Tools / Math Review 1 Resultant Vector...

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