1 angle.mcd Inner product and angle between two vectors. Note that a function is a special case of a vector. Instructor: Nam Sun Wang. x.
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1 angle.mcd Inner product and angle between two vectors. Note that a function is a special case of a vector. Instructor: Nam Sun Wang Define angle between two vectors & y:. y. y. cos( ) (, y). y. y Projection of into y is the inner product of (,y) in the direction of y. (. y). y projection(, y) ( y ) Eample: y Mathcad note: there is a "." after, so that we can do symbolic evaluation later on this worksheet., y deg Schwarz inequality:. y. y "" for "true" Triangular inequality: y y projection, y Eample: sin(. ) sin(. ) projection, y projection, y.577. ( f,. mag( ( f,... Thus, for sine & cosine functions, the angle between the functions vectors coincides with the phase angle. Cross product (torque radius force) torque( r, r F ( r, An eample: asin r r. F F r F torque( r, ( r, deg
2 angle.mcd Define inner product between two functions (and the definition of magnitude naturally follows) An eample:. ( f, f ( ) f ( ) f 3 ( ) f 4 ( ) 3. mag( f, f 9 deg f, f deg f, f 4 9 deg & are not orthogonal. f, f 3 9 deg f, f deg & 3 are not orthogonal. f 3, f 4 9 deg Gram-Schmidt Process. Problem Statement: Given power series f j, construct orthogonal vectors/functions e j. Below ee j are the analytical results. e ( ) f ( ) e ( ) ee ( ) prod e, f prod e e ( ) f ( )., f e ( )... o.k. prod e, e prod e, e e ( ) ee ( ) Mathcad got overwhelmed between e and e 3 and subsequently failed to calculate e 3 and e 4 correctly. prod e, f prod e, f 3 prod e, f prod e, f prod e, prod e, e prod e, e e prod e, e The ratio of the inner product in the second term should have been, not : prod ee, f 3 prod ee, ee.667 prod e, f 3 prod e e 3 ( ) f 3 ( )., f 3 e ( ). e ( ) prod e, e prod e, e prod ee, f 3 prod ee, ee e 3 ( ) 3 ee 3 ( ) 3 prod e, f 4 e 4 ( ) f 4 ( ). e ( ) prod e, e prod e, f 4. e ( ) prod e, e prod e 3, f 4. e 3 ( ) prod e 3, e 3 e 4 ( ) ee 4 ( )
3 , angle.mcd Comparison of numerical results and analytical results. e ( ) ee ( ) e ( ) e 3 ( ) ee ( ) ee 3 ( ) e 4 ( ) ee 4 ( ) Check: (Numerical results are no good.).5.5 e, e 9 deg e, e deg e, e i deg Check: (Analytical results are o.k.) e, e deg e, e i deg e 3, e i deg ee, ee 9 deg ee, ee 3 9 deg ee, ee 4 9 deg ee, ee 3 9 deg ee, ee 4 9 deg ee 3, ee 4 Orthogonal polynomials (Legendre Polynomials.. un-normalized): P ( ) P ( ) P ( ).. 3 P 3 ( ). 9 deg P 4 ( ). 8 P, P 9 deg P, P 9 deg P, P 3 9 deg P, P 4 9 deg P, P 9 deg P, P 3 9 deg P, P 4 9 deg Generating equation for Legendre polynomials:. n P(, n) if n,,.. P(, ) n n n. n P, P 3 9 deg P, P 4 9 deg P(, n ) P 3, P deg The above is a recursive definition that does not work in version 5, but may work in version 6 (provided that we take care of n). Legendre Polynomials of the Second Kind.. not quite mutually orthogonal based on the preceding definition of inner product: Q ( ) ln Q ( ). Q ( ) Q ( ) P. 3 3 ( ) Q ( ). Q, Q 9 deg Q, Q deg Q, Q 7.86 deg
4 4 angle.mcd For certain applications, a different interval is useful. ( f, An eample:... mag( sin( ) cos( ) ( f, 9 deg mag( ) sin( ) sin(. ) ( f, sin( ) sin( 3. ) ( f, cos( ) cos(. ) ( f, 9 deg 9 deg 9 deg sin(. ) cos( 3. ) ( f, 9 deg sin( ) sin( ) sin sin 4 g ( f, 9 deg Another interpretation of angle ( f, 45 deg Another interpretation of angle Thus, sine and cosine functions have the nice property that they are orthogonal to each other. For other applications, a weighting factor may be included. w( ) ( f, w( )... mag( Eample: Chebychev Polynomials. T ( ) T ( ) T ( ). T 3 ( ) T 4 ( ) T, T 9 deg T, T 9 deg T, T 3 9 deg T, T 4 9 deg T, T 9 deg T, T 3 9 deg T, T 4 9 deg T, T 3 9 deg T, T 4 9 deg T 3, T 4 9 deg
5 5 angle.mcd Yet for other applications, a semi-infinite interval is useful, and a weighting factor may be included. w( ) ep( ) ( f, w( )... mag( Eample: Laguerre polynomials. L ( ) L ( ) L ( ) 4. L 3 ( ) L 4 ( ) Although the integrals do not converge numerically (because the upper intergration limit is ), they converge to symbolically. ep( ). L. ( ) L ( ) not converging The angles, calculated symbolically, are: L, L. L, L ep( ). L. ( ) L ( ) ep( ). L. ( ) L ( )., L L 3 L, L. L, L 3., L L 4., L L 4 L, L. 3 L, L 4... L 3, L. 4
6 6 angle.mcd Yet for other applications, an infinite interval is useful, and a weighting factor may be included. w( ) ep ( f, w( )... mag( Eample: Hermite polynomials. H ( ) H ( ). H ( ) 4. H 3 ( ) H 4 ( ) Although the integrals do not converge numerically (because the upper intergration limit is ), they converge to symbolically. overflow ep z. H. ( z) H ( z) dz The angles, calculated symbolically, are: H, H. H, H ep z. H. ( z) H ( z) dz ep z. H. ( z) H ( z) dz., H H 3 H, H. H, H 3., H H 4., H H 4 H, H. 3 H, H 4... H 3, H. 4
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