Engineering Calculus Notes 93

Engineering Calculus Notes 93 - x y z b b b A B C igure...

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1.6. CROSS PRODUCTS 81 plane yields a viewpoint from which the motion appears counterclockwise. We can think of this as replacing the sign σ ( A,B,C ) with a unit vector , normal to the plane containing the three points and pointing toward the side of this plane from which the motion described by our order appears counterclockwise. One way to determine which of the two unit normals is correct is the right-hand rule : point the Fngers of your right hand along the direction of motion; then your (right) thumb will point in the appropriate direction. In ±igure 1.41 we sketch the triangle with vertices
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Unformatted text preview: x y z b b b A B C igure 1.41: Oriented Triangle in R 3 A (2 , 3 , 4), B (4 , 2 , 5), and C (3 , 1 , 3); from our point of view (we are looking from moderately high in the Frst octant), the orientation appears counterclockwise. By interpreting ( A,B,C ) as a unit normal vector, we associate to an oriented triangle ABC R 3 an oriented area v A ( ABC ) = ( A,B,C ) A ( ABC ) represented by a vector normal to the triangle whose length is the ordinary area of ABC . Note that for a triangle in the xy-plane, this means...
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