Lec5-Cross Product

Lec5-Cross Product - z component || u || = < 0,...

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Cross Product, Planes in R 3 Math 32 A Lecture for Wednesday, October 7
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Determinants 2 x 2 3 x 3
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Cross Product
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Cross Product Formula is useful for computations
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Cross Product Formula is useful for computations But what does the cross product mean?
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Right-handed System { v, w, u }
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Geo. Properties of Cross Product v x w is orthogonal to v and w || v x w || = || v || || w || sin( ) v, w, v x w is a right-handed system r
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Geo. Picture of Cross Product r
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Find u = v x w
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u = < 0, a, b> BECAUSE u is orthogonal to v u . v = < 2, 0, 0> . < 0, a, b> = 0
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u = < 0, a, b> = < 0, a, -a> BECAUSE u is orthogonal to w u . w = < 0, 1, 1> . < 0, a, -a> = 0
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u = < 0, a, -a> has length || v || || w || Sin[Pi/2] = 2 Sqrt[2]
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|| u || = < 0, a, -a> = 2 Sqrt[2] = |a| Sqrt[2]
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|| u || = < 0, a, -a> = 2 Sqrt[2] = |a| Sqrt[2] OBSERVE: u has Positive
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Unformatted text preview: z component || u || = &lt; 0, -2, 2&gt; its orthogonal to v and w it has the right length: 2 Sqrt[2] u has positive z component Algebraic Properties Cross Product is NOT commutative! Algebraic Properties It is anti-commutative v x w = - w x v Algebraic Properties Anti-Commutative: because of right-hand rule v x w = - w x v Basic Unit Vectors i x j = k, j x k = i, k x i = j Algebraic Properties Cross Product is NOT associative! Algebraic Properties NOT associative! Example (i x i) x j i x (i x j) Algebraic Properties NOT associative! Example (i x i) x j i x (i x j) PROOF: (i x i) x j = 0 i x (i x j) = i x k = -j Important Fact ||v x w|| = area of parallelogram Important Fact ||u . (v x w)|| = volume of parallelepiped...
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Lec5-Cross Product - z component || u || = &amp;amp;lt; 0,...

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