VECTORS, TENSORS, AND MATRICES. 2 + Az. A vector A can be defined by its length A and the direction of a unit

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1 GG33 Lecture 7 5/17/6 1 VECTORS, TENSORS, ND MTRICES I Min Topics C Vector length nd direction Vector Products Tensor nottion vs. mtrix nottion II Vector Products Vector length: x 2 + y 2 + z 2 vector cn e defined y its length nd the direction of unit vector tht is in the sme direction s. The unit vector hs x,y,z components x i, y j, nd z, respectively, where i,j, nd re unit vectors long the x,y, nd z xes, respectively.. C Exmple: If i + 3j + 4, then , nd 5 i j II Products of Vectors Dot product: M 1 nd re vectors, nd M is sclr corresponding to length. 2 If unit vectors nd prllel vectors nd, respectively, nd the ngle from to (nd from to ) is θ, then, reclling tht cosθ cos( θ ) cos( θ ) cos( θ) cosθ ( ) ( cosθ) c Exmple: If 2i + j + K, nd i + 2j + K, ( 2) ( 2)cos( 9 o ) Stephen Mrtel 7-1 University of Hwii

2 GG33 Lecture 7 5/17/6 2 3 If is unit vector, then (or ) is the length of the projection of onto the direction defined y. 4 Dot product tles of Crtesin sis vectors i j i j i 1 x i x j 1 y j y 1 z z 5 x i + y j + z i + j + x + y + z 6 For unit vectors e r nd e s long xes of Crtesin frme e r e s 1 if r s e r e s if r s 7 In Mtl, C is performed s C(:) *(:) or Csum(.*) 8 Uses in geology for dot products: ll inds of projections Cross product: C 1 C is vector perpendiculr to oth nd, so C is perpendiculr to the plne contining nd. C points in the direction of our thum if the other fingers on your right hnd first point in the direction of nd then curl to point in the direction of. (i.e.,,, nd C form right-hnded set). s result,. Stephen Mrtel 7-2 University of Hwii

3 GG33 Lecture 7 5/17/6 3 2 If unit vectors nd prllel vectors nd, respectively, nd the ngle etween nd (nd etween nd ) is θ, then sinθ n, where n is unit vector norml to the, plne ( ) ( sinθ)n c Exmple: If 2i + j +, nd i + 2j +, ( 2) ( 2)sin( 9 o ) 4 3 The length (mgnitude) of C is the re of the prllelogrm defined y vectors nd, where nd re long djcent side of the prllelogrm. In the figure elow, x points into the pge, nd x points out of the pge. 4 Cross product tles of Crtesin sis vectors i j i j i j x i x x j j i y j y y i j i z z j z i 5 i + j + x y z i + j + y z z y i ( x z )j + x y y x Stephen Mrtel 7-3 University of Hwii

GG33 Lecture 7 5/17/6 4 6 i j x y z 7 For unit vectors e r nd e s long xes of Crtesin frme e p e r if p,q 1,2 or 2,3 or 3,1 e r e p if r,q 3,2 or 2,1 or 1,3 e p if p q 8 In Mtl, C is performed s

4 GG33 Lecture 7 5/17/6 4 6 i j x y z 7 For unit vectors e r nd e s long xes of Crtesin frme e p e r if p,q 1,2 or 2,3 or 3,1 e r e p if r,q 3,2 or 2,1 or 1,3 e p if p q 8 In Mtl, C is performed s Ccross(,) 9 Uses in geology for cross products: finding poles to plnes in threepoint prolems; finding fold xes from poles to edding. C Sclr triple product: (,, C) ( C) V 1 The vector triple product is sclr (i.e., numer) tht corresponds to volume. 2 V is the volume of prllelepiped with edges long,, nd C. (xc) gives the re of the se, nd the dot product of this with gives the se times the component of norml to the se (i.e., the se times the height). The solute vlue of V gurntees tht the volume is non-negtive. Stephen Mrtel 7-4 University of Hwii

5 GG33 Lecture 7 5/17/6 5 3 V ( C) i j C y x y z C y C C x y z z y ( ) + y C C z x y y x 4 The determinnt of 3x3 mtrix gives the volume of prllelepiped. 5 In Mtl, V ( C) is performed s V sum(.*cross(,c)) 6 Use in geology: solutions of equtions, estimting volume of ore odies 7 If t lest two of the vectors,,c re prllel to ech other, then,,c cnnot define prllelepiped, t lest two rows of the mtrix in (3) re linerly dependent, nd the determinnt of (3) is zero, nd the three plnes defined y,,c will not intersect in unique point 8 Proof tht C is perpendiculr to the plne of nd If C is not perpendiculr to the plne, then C must e nonperpendiculr to oth nd C, i.e., C nd. C ( ) ( ) c C ( ) ( ) d The postulte tht C is not perpendiculr to the plne thus is disproved, so C is perpendiculr to the plne. D Invrints 1 Quntities tht do not depend on the orienttion of coordinte system. 2 Exmples Dot product of two vectors ( length) Sclr triple product ( volume) Stephen Mrtel 7-5 University of Hwii

Things to Memorize: A Partial List. January 27, 2017

Things to Memorize: A Partial List. January 27, 2017 Things to Memorize: A Prtil List Jnury 27, 2017 Chpter 2 Vectors - Bsic Fcts A vector hs mgnitude (lso clled size/length/norm) nd direction. It does not hve fixed position, so the sme vector cn e moved