T3 - 4. For a problem in finding a vector, one good method...

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ERG 2018 Tutorial 3 September 23, 2008 EX 1 Some useful equalities and inequalities for vectors. Suppose we have vectors ~a and ~ b , both in the space of R n . Triangle rule k ~a k + k ~ b k ≥ k ---→ a - b k Triangle rule k ---→ a - b k ≥ k ~a k - k ~ b k Inner Product ~a · ~ b = k ~a kk ~ b k cos θ Cosine Law k ---→ a - b k 2 = k ~a k 2 + k ~ b k 2 - 2 k ~a kk ~ b k cos θ Sine Law k ~a k sin A = k ~ b k sin B where θ is the angle between ~a and ~ b , A is the angle between ~ b and ---→ b - a , and B is the angle between ~a and ---→ a - b . To show the Sine and Cosine laws, you should give a way to handle the angles calculated by vectors. EX 2 A nonzero vector ~x = ( x 1 ,x 2 ,...,x n ) can be written in the form of x = a · ~ x 0 , where k ~ x 0 k = 1 and a is a positive real number. a is also called the length of the vector, and ~ x 0 is a unit vector parallel to ~x . 1. For ~x = (1 , 2 , 2), find the length of ~x and the unit vector parallel to ~x . Do the same work for ~ y = ( - 1 , - 2 , - 2) and ~ z = (2 , 4 , 4) 2. Decide whether the above vectors are parallel to each other. Could you find the parallel relation within the scalar times unit vector form a · ~ x 0 ? 3. Find the unit vector which is orthogonal to the vectors above. Is it unique?
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Unformatted text preview: 4. For a problem in finding a vector, one good method is to find its length and the unit vector parallel to it? Given vectors ~a = [1 , , 1] T and ~ b = [1 ,-2 , 2] T , what is the vector when ~a is projected on ~ b ? 1 EX 3 Given a set of vectors { ~ v 1 , ~ v 2 ,..., ~ v n } , consider the following system: x 1 ~ v 1 + x 2 ~ v n + ··· + x n ~ v n = ~ If there is only zeros solution to this system, we call the set of vectors are Linearly Independent , otherwise we call they are Linearly Dependent . If there are LI, x 1 ~ v 1 + x 2 ~ v n + ··· + x n ~ v n is the vector space spanned by these vectors. How about the geometric view of the spanned space? We should work more. Let ~a and ~ b be defined in the EX2. Find the vector space spanned by these vectors using Gauss elimination. 2...
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This note was uploaded on 11/13/2010 for the course SEEM SEEM2018 taught by Professor Chan during the Spring '10 term at CUHK.

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T3 - 4. For a problem in finding a vector, one good method...

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