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

This preview shows pages 1–2. Sign up to view the full content.

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?

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## This note was uploaded on 11/13/2010 for the course SEEM SEEM2018 taught by Professor Chan during the Spring '10 term at CUHK.

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online