crossproduct

crossproduct - Cross Product Given two vectors u and v ,...

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Unformatted text preview: Cross Product Given two vectors u and v , the cross product u × v between them is defined as: 1. Analytic Definition: Let u = (x1 , y1 , z1 ) and v = (x2 , y2 , z2 ). Then u × v = (y 1 z2 − z1 y 2 , z1 x2 − x1 z2 , x 1 y2 − y1 x2 ) 2. Geometric Definition: u × v is a vector such that: its length u × v equals u v sin θ where 0 ≤ θ ≤ π is the angle between them; and its direction is determined by the right-hand-rule: fingers=u , palm=v , and thumb=u × v . Note: given two vectors, the cross product results in a vector. Properties of Cross Product u×v u × (u + w ) (u + w ) × u αu × v O ×u u · (v × w ) u × (v × w ) = = = = = = = −v × u u×v +u×w v ×u+w ×u u × αv O (u × v ) · w (u · w )v − (u · v )w Note: u × (v × w ) = (u × v ) × w . Note: ˆ × ˆ = k , ˆ × k = ˆ, k × ˆ = ˆ i j ˆj ˆ iˆ i j ˆˆ j and ˆ × ˆ = −k , k × ˆ = −ˆ, ˆ × k = −ˆ. ji ii ˆ j Additional Properties of Cross Product 1. u × u = O . 2. two (non-zero) vectors are parallel to each other (i.e. with the same or opposite direction or θ = 0, π ) if and only if their cross product equals the zero vector. 3. u × v is perpendicular to both u and v . 4. u × v equals the area of the parallelogram spanned by u and v or 1 u × v equals the 2 area of the triangle spanned by u and v . 5. |u · (v × w )| (absolute value of the scalar triple product) equals the volume of the parallelepiped spanned by u , v and w . ...
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