lecture2-09[1] - 18.02 Multivariable Calculus(Spring 2009...

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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 using the coordinates of the vectors, that is if A = ( a 1 , a 2 , a 3 ) , B = ( b 1 , b 2 , b 3 ) , are two vectors in 3D forming an angle θ , then A · B = | A || B | cos θ = a 1 b 1 + a 2 b 2 + a 3 b 3 . 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 n where ˆ n is the unit vector to both A and B that satisfies the Right Hand Rule (RHR) 1 (figure relative to cross product) 1 Think about a screw! 1
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Handy Facts 1. | A × B | = | A || B | sin θ = area of parallelogram spanned by the vectors A and B .
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