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Properties of Vector Addition 1. Commutative Property of Addition: a + b = b + a 2. Associative Property of Addition: (a + b) + 7 = 7 + (b + ?) 3. Distributive Property of Addition: k(a + b) = ka + kb, keR Eg.1: Simplify the following expression: 4 3a - 6+2c - 2a-46+3c Further Laws of Vector Addition and Scalar Multiplication 1. Adding O: a + 0 = a 2. Associative Law for Scalars: m(nd) = (mn)d = mnd 3. Distributive Law for Scalars: (m + n)d = ma + nd Eg.2: If x = 21-3j+4k, y=5j-2k and z= - i-2 j+3k , then determine each of the following in terms of i , j and k . a) xty b) x - y x - 2y+ 3z Homework: p. 307 # 5 - 10 , 11**A linear combination occurs when vectors are multiplied by scalars and then added, the result is a new vector that is a linear combination of the vectors. If we consider the vectors a = [3,-1] and b = [-4, 2] and write 2[3, -1]- 3[-4, 2] = [18, -8] , then the expression on the left side of this equation is a linear combination or the right side. Spanning Set for R2 The set of vectors /. j is said to form a spanning set for R-. Every vector in R- can be written uniquely as a linear combination of these two vectors. Writing Vectors as Linear Combinations of two other vectors Eg.1: Write q = 4, -7 as a linear combination of u = [2, -1] and v = [0,5]Showinn that a given set of vectors spans R2 Eg.2: Show that the set of vectors {[2, Hand [3, 1]} is a spanning set of R2. Recall: Collinear vectors are scalar multiples of one another. Coplanar vectors are vectors that lie on the same plane. If 3 vectors are coplanar, then the 3 vectors can be expressed as linear combinations of each other. E33: Determine if r; = [5,0,6] and 3: [3,2,0]and E: [6,1,9]are coplanar. Linear Dependence: Vectors are considered to be linearly dependent if they are collinear in 2-space and coplanar in 3-space. Otherwise they are linearly independent