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Rog_Sec_12_4 - Math 20C Lecture Examples Section 12.4 The...

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(8/1/08) Math 20C. Lecture Examples. Section 12.4. The cross product There are two directions perpendicular to two nonzero and nonparallel vectors v and w in xyz -space. They are distinguished by using the right-hand rule : A nonzero vector u perpendicular to v and w has the direction given by the right-hand rule from v toward w if, when the fingers of a right hand curl from u toward v , as in Figure 1, the thumb points in the direction of u . FIGURE 1 FIGURE 2 The right-hand rule is used in the definition of the cross product of two vectors. Definition 1 The cross product v × w of nonzero and nonparallel vectors v and w in xyz -space is the vector perpendicular to v and w with direction determined by the right-hand rule from v toward w and whose length is | v × w | = | v || w | sin θ (1) where θ is the angle with 0 < θ < π between v and w . (Figure 2). If v or w is the zero vector or they are parallel, then v × w is the zero vector. The cross product has the properties listed in the next theorem. Notice the minus sign in equation (2) . Theorem 1 For any vectors u , v , and w in xyz -space and any number λ , v × w = - w × v (2) ( λ v ) × w = v × ( λ w ) = λ ( v × w ) (3) u × ( v + w ) = u × v + u × w (4) . Lecture notes to accompany Section 12.4 of Calculus, Early Transcendentals by Rogawski. 1
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Math 20C. Lecture Examples. (8/1/08) Section 12.4, p. 2 Calculating cross products with determinants Cross products can be calculated using the notation of determinants from linear algebra. The 2 × 2 (two-by-two) determinant vextendsingle vextendsingle vextendsingle vextendsingle x 1 x 2 y 1 y 2 vextendsingle vextendsingle vextendsingle vextendsingle
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