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la_m5_handouts

# la_m5_handouts - MAC 2103 Module 5 Vectors in 2-Space and...

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1 1 MAC 2103 Module 5 Vectors in 2-Space and 3-Space II 2 Rev.F09 Learning Objectives Upon completing this module, you should be able to: 1. Determine the cross product of a vector in 3 . 2. Determine a scalar triple product of three vectors in 3 . 3. Find the area of a parallelogram and the volume of a parallelepiped in 3 . 4. Find the sine of the angle between two vectors in 3 . 5. Find the equation of a plane in 3 . 6. Find the parametric equations of a line in 3 . 7. Find the distance between a point and a plane in 3 . http://faculty.valenciacc.edu/ashaw/ Click link to download other modules.

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2 3 Rev.09 Vectors in 2-Space and 3-Space II http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Cross Products Cross Products Lines and Planes in 3-Space Lines and Planes in 3-Space There are two major topics in this module: 4 Rev.F09 Quick Review: The Norm of a Vector in 3 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. The norm of a vector u, , is the length or the magnitude of the vector u . • If u = ( u 1 , u 2 , u 3 ) = (-1, 4, -8), then the norm of the vector u is This is just the distance of the terminal point to the origin for u in standard position. Note: If u is any nonzero vector, then is a unit vector. A unit vector is a vector of norm 1. ! u = u ! u = u 1 2 + u 2 2 + u 3 2 = ( ! 1) 2 + 4 2 + ( ! 8) 2 = 9 u ! u
3 5 Rev.F09 The Cross Product of Two Vectors in 3 http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. The cross product of two vectors u = ( u 1 ,u 2 ,u 3 ) and v = ( v 1 ,v 2 ,v 3 ), u x v, in 3 is a vector in 3 . The direction of the cross product , u x v , is always perpendicular to the two vectors u and v and the plane determined by u and v that is parallel to both u and v . The norm of the cross product is u x v u v ! u ! ! v = ! u ! v sin( " ). 6 Rev.F09 The Cross Product of Two Vectors in 3 (Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. The cross product can be represented symbolically in the form of a 3 x 3 determinant: u x v = where i = (1,0,0), j = (0,1,0), k = (0,0,1) are standard unit vectors . i j k u 1 u 2 u 3 v 1 v 2 v 3 = u 2 u 3 v 2 v 3 i ! u 1 u 3 v 1 v 3 j + u 1 u 2 v 1 v 2 k = u 2 u 3 v 2 v 3 , ! u 1 u 3 v 1 v 3 , u 1 u 2 v 1 v 2 " # \$ \$ % & ' ' Note: Every vector in 3 is expressible in terms of the standard unit vectors. v = ( v 1 ,v 2 ,v 3 ) = v 1 (1,0,0) + v 2 (0,1,0) + v 3 (0,0,1) = v 1 i + v 2 j + v 3 k

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4 7 Rev.F09 The Cross Product of Two Vectors in 3 (Cont.) http://faculty.valenciacc.edu/ashaw/ Click link to download other modules. Example: Find the cross product of u = (0,2,-3) and v = (2,6,7).
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