lec_week2

# lec_week2 - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 18.02 Lecture 3. – Tue, Sept 11, 2007 Remark: A × B = − B × A , A × A = 0. Application of cross product : equation of plane through P 1 ,P 2 ,P 3 : P = ( x,y,z ) is in the plane iff det( −− P 1 P , P −−− 1 P → 2 , −−−→ P 1 P N = 0, where N is the normal vector P 1 P 3 ) = 0, or equivalently, −−→ · N = −−− P 1 P → 2 × −−−→ I explained this geometrically, and showed how we get the same equation both P 1 P 3 . ways. Matrices. Often quantities are related by linear transformations; e.g. changing coordinate systems, from P = ( x 1 ,x 2 ,x 3 ) to something more adapted to the problem, with new coordinates ( u 1 ,u 2 ,u 3 ). For example ⎧ ⎨ u 1 = 2 x 1 + 3 x 2 + 3 x 3 u 2 = 2 x 1 + 4 x 2 + 5 x 3 ⎩ u 3 = x 1 + x 2 + 2 x 3 ⎡ ⎤⎡ ⎤ ⎡ ⎤ 2 3 3 x 1 u 1 Rewrite using matrix product: ⎣ 2 4 5 ⎦⎣ x 2 ⎦ = ⎣ u 2 ⎦ , i.e. AX = U . 1 1 2 x 3 u 3 Entries in the matrix product = dot product between rows of A and columns of X . (here we multiply a 3x3 matrix by a column vector = 3x1 matrix). More generally, matrix multiplication AB : ⎡ ⎤ ⎡ ⎤ . ⎡ ⎤ 1 2 3 4 ⎢ . 3 ⎥ . 14 ⎣ . . . . ⎦⎢ ⎣ . ⎥ ⎦ = ⎣ . . ⎦ . . . . . . . 2 (Also explained one can set up A to the left, B to the top, then each entry of AB = dot product between row to its left and column above it). Note: for this to make sense, width of A must equal height of B . What AB means: BX = apply transformation B to vector X , so ( AB ) X = A ( BX ) = apply first B then A . (so matrix multiplication is like composing transformations, but from right to left!) (Remark: matrix product is not commutative, AB is in general not the same as BA – one of the two need not even make sense if sizes not compatible)....
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lec_week2 - MIT OpenCourseWare http/ocw.mit.edu 18.02...

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