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MIT 18 02 - Lecture 2 Deteminants

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18.02 Multivariable Calculus (Spring 2009): Lecture 2Deteminants. Cross Product.February 5Reading Material: From Simmons: 18.3. From Course Notes D.Last time: Vectors. Vector arithmetic. Dot product.Today: Cross product. Determinant.2 Cross ProductYesterday in your recitation you learned that the dot product of two vectors can be also expressedusing the coordinates of the vectors, that is if~A = (a1, a2, a3),~B = (b1, b2, b3), are two vectors in 3Dforming an angle θ, then~A ·~B = |~A||~B| cos θ = a1b1+ a2b2+ a3b3.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.Definition 1. The Cross Product of two vectors~A and~B (only for 3D) is defined as~A ×~B = (|~A||~B| sin θ)ˆnwhere ˆn is the unit vector ⊥ to both~A and~B that satisfies the Right Hand Rule (RHR)1(figure relative to cross product)1Think about a screw!1Handy Facts1. |~A ×~B| = |~A||~B| sin θ = area of parallelogram spanned by the vectors~A and~B.2.~A ×~B by construction gives a vector orthogonal both to~A and~B. This will be very importantin the future!To compute~A ×~B with coordinates we need a new mathematical tool: determinant.3 DeterminantsDefinition 2. A m by n (m × n) matrix A is a table of scalarsa11a12· · · a1na21. . .am1amnThis matrix has n columns and m rows. If n = m we say that the matrix is square. If the matrixis square (m = n) then it has a magic # called determinant. We denote this number bydet A or |A|.In the 1 × 1 case A = (a) for some scalar a and we simply havedet A = a.In the 2 × 2 caseA =a1a2b1b2we define det A asdet A = a1b2− a2b1.Exercise 1.1 23 4=In the 3 × 3 caseA =a1a2a3b1b2b3c1c2c3thendet A = a1b2c3+ a2b3c1+ a3b1c2− a3b2c1− a2b1c3− a1b3c2Exercise 2.1 2 30 7 11 9 4=2Handy FactsTo calculate determinants the following facts are quite useful:1. Exchanging 2 rows → det flips sign2. 2 (or more) identical rows in the matrix → det = 03. add/subtract a row from another → no change in detSame considerations for columns.Remark. You will see in recitation that determinants are useful in order to calculate volumes. Infact you will see that ifA =a1a2a3b1b2b3c1c2c3~A = (a1, a2, a3),~B = (b1, b2, b3) and~C = (c1, c2, c3), then| det A| = Volume of parallelogram spanned by~A,~B and~C.Another calculation method: cofactor expansion methodHere I am going to compute the determinant of a 3 × 3 matrix by a cofactor expansion relative tothe first row:a1a2a3b1b2b3c1c2c3= a1b2b3c2c3− a2b1b3c1c3+ a3b1b2c1c2= a11m11− a12m12+ a13m13= a11c11+ a12c12+ a13c13.wheremij= i, j minor = det after remove row i col jcij= i, j cofactor = (−1)i+jmij3Once the minors with respect to a certain row have been found then to compute the cofactors onejust needs to remember the following distribution of signs on a 3 × 3 matrix:+ − +− + −+ − +Theorem 1. Computing~A ×~BIf~A = (a1, a2, a3) and~B = (b1, b2, b3) then~A ×~B =ˆiˆjˆka1a2a3b1b2b3.Exercise 3. Given two vectors~A = (3, −2, 4) and~B = (2, 1, −2) find a vector~N ⊥ to both~A and~B.4Study 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. Arethe two vectors in Ex. 3 parallel? If I had given two parallel vectors in Ex. 3, for example~A = (0, −2, 4) and~C = (0, −1, 2), what would have happened? Think about this question bothin a geometric way (using a picture) and in an analytic way by computing the


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MIT 18 02 - Lecture 2 Deteminants

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