Useful Formulas for Final

# Useful Formulas for Final - Let a = a1 a2 a3 and b = b1 b2...

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Let a = h a 1 ,a 2 ,a 3 i and b = h b 1 ,b 2 ,b 3 i be vectors and θ the angle between them. Length: | a | = p a 2 1 + a 2 2 + a 2 3 . Dot product: a · b = a 1 b 1 + a 2 b 2 + a 3 b 3 = | a || b | cos θ . a is parallel to b if a = c b , for some scalar c . a and b are perpendicular if a · b = 0. Vector projection of b onto a : proj a ( b ) = comp a ( b ) u , where u = a / | a | is a unit vector and comp a ( b ) = b · u is the component of b in the direction of a . Cross product: a × b = h a 2 b 3 - a 3 b 2 , a 3 b 1 - a 1 b 3 , a 1 b 2 - a 2 b 1 i , if ± ± ± ± a b c d ± ± ± ± = ad - bc a × b = ± ± ± ± ± ± i j k a 1 a 2 a 3 b 1 b 2 b 3 ± ± ± ± ± ± = ± ± ± ± a 2 a 3 b 2 b 3 ± ± ± ± i - ± ± ± ± a 1 a 3 b 1 b 3 ± ± ± ± j + ± ± ± ± a 1 a 2 b 1 b 2 ± ± ± ± k , | a × b | = | a || b | sin θ , a × b is perpendicular to both a and b . The area of the triangle with a and b as two edges is A = | a × b | / 2. The volume of the parallelepiped with a , b and c as three edges is V = | a · ( b × c ) | . a × b = 0 if a and b are parallel. a · ( b × c ) = 0 if a , b and c lie in the same plane. Parametric equations of a line: x = x 0 + at , y = y 0 + bt , z = z 0 + ct where ( x 0 ,y 0 ,z 0 ) is on the line and h a,b,c i is parallel to it:

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Useful Formulas for Final - Let a = a1 a2 a3 and b = b1 b2...

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