MATH136 Lecture Notes - Lecture 7: If And Only If, Linear Algebra

• School: University of Waterloo
• Department: Mathematics
• Course: MATH136
• Professor: Alejandra Vicente Colmenares

Document Summary

Math 136 lecture 7 17 jan, 2018. In linear algebra we often need to find a vector that is orthogonal to some other vectors. For example, how would we find a vector (cid:2871) that is orthogonal to two given vectors (cid:1874)(cid:2869) ,(cid:1874)(cid:2870) (cid:2871). There are infinitely many solution, we typically pick the special solution: (cid:882)= (cid:1874)(cid:2869) =(cid:2869)(cid:1874)(cid:2869)(cid:2869)+ (cid:2870)(cid:1874)(cid:2869)(cid:2870)+(cid:2871)(cid:1874)(cid:2869)(cid:2871) (cid:882)= (cid:1874)(cid:2870) =(cid:2869)(cid:1874)(cid:2870)(cid:2869)+ (cid:2870)(cid:1874)(cid:2870)(cid:2870)+(cid:2871)(cid:1874)(cid:2870)(cid:2871) Find a vector orthogonal to (cid:1873) and (cid:1874) Let (cid:1874)(cid:2869) =[(cid:1874)(cid:2869)(cid:2869)(cid:1874)(cid:2869)(cid:2870)(cid:1874)(cid:2869)(cid:2871)],(cid:1874)(cid:2870) =[(cid:1874)(cid:2870)(cid:2869)(cid:1874)(cid:2870)(cid:2870)(cid:1874)(cid:2870)(cid:2871)], then =[(cid:1874)(cid:2869)(cid:2870)(cid:1874)(cid:2870)(cid:2871) (cid:1874)(cid:2869)(cid:2871)(cid:1874)(cid:2870)(cid:2870) (cid:1874)(cid:2869)(cid:2871)(cid:1874)(cid:2870)(cid:2869) (cid:1874)(cid:2869)(cid:2869)(cid:1874)(cid:2870)(cid:2871) (cid:1874)(cid:2869)(cid:2869)(cid:1874)(cid:2870)(cid:2870) (cid:1874)(cid:2870)(cid:2869)(cid:1874)(cid:2869)(cid:2870)] is called the cross-product of (cid:1874)(cid:2869) and (cid:1874)(cid:2870) and is denoted by =(cid:1874)(cid:2869) (cid:1874)(cid:2870) . The formula works only for vectors in (cid:2871). For (cid:2872), 6, , make a linear equation for those vectors, i. e. (cid:1874)(cid:2869) =(cid:882), (cid:1874)(cid:2870) =(cid:882), . Theorem 1. 4. 5: (cid:1874) ,(cid:1877) ,(cid:1878) (cid:2871),(cid:1871) , then: if =(cid:1874) (cid:1877) , then for any (cid:1878) =(cid:1871)(cid:1853){(cid:1874) ,(cid:1877) }, we have (cid:1878) =(cid:882, (cid:1874) (cid:1877) = (cid:1877) (cid:1874, (cid:1874) (cid:1874) =(cid:882)