18 02 Multivariable Calculus Spring 2009 Lecture 2 Deteminants Cross Product February 5 Reading Material From Simmons 18 3 From Course Notes D Last time Vectors Vector arithmetic Dot product Today Cross product Determinant 2 Cross Product Yesterday in your recitation you learned that the dot product of two vectors can be also expressed a1 a2 a3 B b1 b2 b3 are two vectors in 3D using the coordinates of the vectors that is if A forming an angle then B A B cos a1 b1 a2 b2 a3 b3 A Observe that there is an equivalent formula in 2D We now introduce a different kind of product of vectors This one can only be defined for 3D vectors and B only for 3D is defined as Definition 1 The Cross Product of two vectors A B A B sin n A and B that satisfies the Right Hand Rule RHR 1 where n is the unit vector to both A figure relative to cross product 1 Think about a screw 1 Handy Facts B A B sin area of parallelogram spanned by the vectors A and B 1 A B by construction gives a vector orthogonal both to A and B This will be very important 2 A in the future B with coordinates we need a new mathematical tool determinant To compute A 3 Determinants Definition 2 A m by n m n matrix A is a table of scalars a11 a12 a1n a21 am1 amn This matrix has n columns and m rows If n m we say that the matrix is square If the matrix is square m n then it has a magic called determinant We denote this number by det A or A In the 1 1 case A a for some scalar a and we simply have det A a In the 2 2 case A a1 b1 a2 b2 we define det A as det A a1 b2 a2 b1 Exercise 1 1 3 2 4 In the 3 3 case a1 A b1 c1 a2 b2 c2 a3 b3 c3 then det A a1 b2 c3 a2 b3 c1 a3 b1 c2 a3 b2 c1 a2 b1 c3 a1 b3 c2 Exercise 2 1 0 1 2 7 9 3 1 4 2 Handy Facts To calculate determinants the following facts are quite useful 1 Exchanging 2 rows det flips sign 2 2 or more identical rows in the matrix det 0 3 add subtract a row from another no change in det Same considerations for columns Remark You will see in recitation that determinants are useful in order to calculate volumes In fact you will see that if a1 a2 a3 A b 1 b2 b3 c1 c2 c3 a1 a2 a3 B b1 b2 b3 and C c1 c2 c3 then A B and C det A Volume of parallelogram spanned by A Another calculation method cofactor expansion method Here I am going to compute the determinant of a 3 3 matrix by a cofactor expansion relative to the first row a1 b1 c1 a2 b2 c2 a3 b3 c3 a1 b2 c2 b3 c3 a2 b1 c1 b3 c3 a3 b1 c1 b2 c2 a11 m11 a12 m12 a13 m13 a11 c11 a12 c12 a13 c13 where mij i j minor det after remove row i col j cij i j cofactor 1 i j mij 3 Once the minors with respect to a certain row have been found then to compute the cofactors one just needs to remember the following distribution of signs on a 3 3 matrix B Theorem 1 Computing A a1 a2 a3 and B b1 b2 b3 then If A B A i a1 b1 j a2 b2 k a3 b3 3 2 4 and B 2 1 2 find a vector N to both A and Exercise 3 Given two vectors A B 4 Study Guide 1 The questions for this lecture are What is the geometric meaning of the cross product Is the determinant defined for rectangular matrices How do you compute the volume of a parallelepiped spanned by three vectors Two vectors are said to be parallel if one is a scalar non zero multiple of the other one Are the two vectors in Ex 3 parallel If I had given two parallel vectors in Ex 3 for example 0 2 4 and C 0 1 2 what would have happened Think about this question both A in a geometric way using a picture and in an analytic way by computing the determinant 5
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