MAT208 FALL 2009Length and Direction in R2 and R3Slide 3AnglesSlide 5Slide 6Direction CosinesInner ProductOrthogonalityUnit VectorsCross ProductSlide 12Cross Product in R3 OrthogonalityProperties of the Cross ProductCross Products of Natural Basis VectorsAngles AgainSlide 17Cross Product in R2 ??Determinants and Cross ProductInner Product SpacesSlide 21Slide 22Slide 23Slide 24Slide 25Positive Definite MatrixDefinitionsCauchy-Schwarz InequalityCauchy-Schwarz (continued)Slide 30The Triangle InequalityThe Triangle Inequality (continued)More DefinitionsSlide 34OrthonormalMAT208 FALL 2009Sections 5.1-5.3Kolman/HillLength and Direction in R2 and R3Let be a vector in R2. The length or magnitude of the vector, denoted by is Also called norm. • Let and be vectors in R2. The distance between u and v is defined as the magnitude of u - v, i.e. as 12vv� �� �� �� �=vv2 21 2v v= +v12uu� �� �� �� �=u12vv� �� �� �� �=v( ) ( )2 21 1 2 2u v u v- = - + -u vLength and Direction in R2 and R3Concepts can be extended to R3 1 12 23 3u vu vu v� � � �� � � �� � � �� � � �� � � �= =u v( ) ( )( )2 2 21 2 322 21 1 2 2 3 3u u uu v u v u v= + +- = - + - + -uu vAnglesFind the angle between two vectors 1 12 23 3u vu vu v� � � �� � � �� � � �� � � �� � � �= =u vuvv uLengths of sides are , and vu-v uAnglesBy the law of cosines 2 2 22 cosq- = + -v u u v u v( ) ( )( )( )2 2 222 22 2 2 2 2 21 2 3 1 2 3 1 1 2 2 3 31 1 2 2 3 3cos22u u u v v v v u v u v uv u v u v uq+ - -=+ + + + - - + - + -=+ +=+u v v uu vu vu vAn angle can be computed from the coordinates of the vectorsAnglesFor vectors and in R2 1 12 2u vu v� � � �� � � �� � � �� � � �= =u v1 1 2 2cosv u v uq+=u vDirection CosinesIn R3, let and letbe the natural basis for R3. Let a be the angle between v and the x-axis, b be the angle between v and the y-axis and c be the angle between v and the z-axis.The quantities cos a, cos b and cos are called the direction cosines of v and can be computed from v and i, j, k . 123vvv� �� �� �� �� �=v1 0 00 , 1 , 00 0 1�� �� ���� �� ���� �� ���� �� ���� �� ��= = =i j kcosv1vcosv2vcosv3vgInner Product Let and be vectors in R2. The inner product or dot product of u and v, denoted by ( u,v ) or u•v, is u•v = u1v1 + u2v2. Similarly, for and in R3, u•v = u1v1 + u2v2 + u3v31 12 2u vu v� � � �� � � �� � � �� � � �= =u v1 12 23 3u vu vu v� � � �� � � �� � � �� � � �� � � �= =u vOrthogonalityTwo vectors u and v in R2 or R3 are orthogonal or perpendicular if u•v = 0.Theorem - Let u, v and w be vectors in R2 or R3. Then 1) u•u > 0 if u ≠ 0, and u•u = 0 if and only if u = 02) u•v = v•u3) (u + v)•w = u•w + v•w4) (c u)•v = c (u•v), for any real scalar cUnit VectorsA unit vector in R2 or R3 is a vector whose length is 1.Note - If x is a nonzero vector, then the vector is a unit vector .u 1xxCross ProductLet u, v R3 .1231 2 3oruuuu u u� �� �� �� �� �== + +uu i j k1231 2 3orvvvv v v� �� �� �� �� �== + +vv i j kTry to find a vector w that is orthogonal to both u and v, i.e. u • w = 0, v • w = 0 xyz� �� �� �� �� �=w1 2 31 2 300u x u y u zv x v y v z+ + =+ + =Cross ProductTwo equations in the three unknowns x, y and z. No unique solution for x, y and z. One solution is x = u2v3 - u3v2, y = u3v1 - u1v3, z = u1v2 - u2v1 .( ) ( )( )2 3 3 23 1 1 31 2 2 12 3 3 2 3 1 1 3 1 2 2 1u v u vu v u vu v u vu v u v u v u v u v u v� �� �� �� �� �-= --= - + - + -= �wi j ku vCross Product in R3OrthogonalityNote that ( ) ( )( )1 2 3 3 2 2 3 1 1 3 3 1 2 2 11 2 3 1 3 2 2 3 1 2 1 3 3 1 2 3 2 1•0u u v u v u u v u v u u v u vu u v u u v u u v u u v u u v u u v= - + - + -= - + - + -=u wSimilar calculation shows that v • w = 0Properties of the Cross Producta) u x v = - v x ub) u x (v + w) = u x v + u x w c) (u + v) x w = u x w + v x wd) c (u x v) = (c u) x v = u x (c v) e) u x u = 0f) 0 x u = u x 0 = 0g) u x (v x w) = (u • w) v - (u • v) wh) (u x v) x w = (w • u) v - (w • v) uCross Products of Natural Basis VectorsCross Products of Natural Basis VectorsDirection of w = u x v determined by right hand rule.�= � = � =�= � = � =�=- �=- � =-i i 0 j j 0 k k 0i j k j k i k i jj i k k j i i k jAngles AgainAn angle between two vectors can also be determined by using the cross product. We will need the following identity: (u x v) • w = u • (v x w)Angles(continued)( ) ( ) ( )( )( ) ( )( )( ) ( ) ( )( )222 2 2 222 2 2 22 2• •• • • • • •cos1 cos sinqq q� = � � = � �= - = -= -= - =u v u v u v u v u vu v v u v u v v v u u v uv u v uv u v usin sinq q�� = � =u vu v u vu vCross Product in R2 ??If u and v are in R2 then define cross product for them by consideringThis is not a vector in R2, but it is still useful:Consider complex …
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