Since a ˆ ı a 1 etc it follows immediately that a a

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Since a · ˆ ı = a 1 etc, it follows immediately that a = a ( λ ˆ ı + µ ˆ + ν ˆ k ) λ 2 + µ 2 + ν 2 = 1 a 2 [ a 2 1 + a 2 2 + a 2 3 ] = 1 i k j
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Vector or cross product 1.22 The vector product of two vectors a and b is a × b = ( a 2 b 3 a 3 b 2 ı + ( a 3 b 1 a 1 b 3 + ( a 1 b 2 a 2 b 1 ) ˆ k . You cannot remember the above! Instead use the pseudo determinant a × b = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ˆ ı ˆ ˆ k a 1 a 2 a 3 b 1 b 2 b 3 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle where the top row consists of vectors not scalars. A determinant with two equal rows has value zero, so a × a = 0 It is also easily verified that ( a × b ) · a = ( a × b ) · b = 0 so that a × b is orthogonal to both a and b .
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Vector product 1.23 The magnitude of the vector product can be obtained by showing that | a × b | 2 + ( a · b ) 2 = a 2 b 2 from which it follows (independent of the coord system) | a × b | = ab sin θ , Proof? The vector product does not commute It anti-commutes : a × b = b × a . The vector product does not associate: a × ( b × c ) negationslash = ( a × b ) × c .
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Vector Products 1.24 The vector product is orthogonal to both the vectors. Need to specify the sense w.r.t these vectors. Sense of the right handed screw ... in right-hand screw sense Plane of vectors a and b a x b b a Also ˆ ı × ˆ = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ˆ ı ˆ ˆ k 1 0 0 0 1 0 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = ˆ k . And in full: ˆ ı × ˆ = ˆ k , ˆ × ˆ k = ˆ ı , and ˆ k × ˆ ı = ˆ . Note the cycle ordering here.
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Geometrical interpretation of vector product 1.25 The magnitude of the vector product ( a × b ) is equal to the area of the parallelogram whose sides are parallel to, and have lengths equal to the magnitudes of, the vectors a and b . Its direction is perpendicular to the parallelogram. θ x a b b bsin θ a
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Example 1.26 Question g is vector from A [1,2,3] to B [3,4,5]. ˆ l is the unit vector in dirn from O to A. Find ˆm , a UNIT vector along g × ˆ l Verify that ˆm is is perpendicular to ˆ l . Find ˆn , the third member of a r-h coord set ˆ l , ˆm , ˆn . g l A B [1,2,3] [3,4,5] Answer 1) g = [3 1 , 4 2 , 5 3] = [2 , 2 , 2]. 2) ˆ l = [1 , 2 , 3] / 14 3) g × l = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ˆ ı ˆ ˆ k 2 2 2 1 2 3 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle = [2 , 4 , 2] ˆm = [1 , 2 , 1] / 6 4) ˆ l · ˆm = (1 . 1 + 2 . 2 + 1 . 3) / ( . ) = 0 5) ˆn = ˆ l × ˆm = 1 6 14 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle ˆ ı ˆ ˆ k 1 2 3 1 2 1 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle
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Random question 1.27 Q: If ˆ f and ˆg are two unit vectors, what is the magnitude of the vector product ˆ f × ˆg f x g g f θ A: Magnitude is sin θ .
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Summary 1.28 We’ve revised and discussed ...
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