At least one of the answers above is NOT correct. Let V = (7,∞). For u, v Є V and a Є R define vector addition by u = v := uv − 7(u + v) + 56 and scalar multiplication by a u := (u − 7) ª + 7. It can be shown that (V, Œ, □) is a vector space over the scalar field R. Find the following: the sum: 8田9= = 9 the scalar multiple: -1 8 = 8 the additive inverse of 8: 日8 63 the zero vector: Ων 8 the additive inverse of x: 1 Ac +7 x-7

Illustrate properties 110 of Theorem 4.2 for u=(2,1,3,6), v=(1,4,0,1), w=(3,0,2,0), c=5, and d=2. THEOREM 4.2Properties of Vector Addition and Scalar Multiplication in Rn. Let u,v, and w be vectors in Rn, and let c and d be scalars. 1. u+v is vector in Rn. Closure under addition 2. u+v=v+u Commutative property of addition 3. (u+v)+w=u+(v+w) Associative property of addition 4. u+0=u Additive identity property 5. u+(u)=0 Additive inverse property 6. cu is a vector in Rn. Closure under scalar multiplication 7. c(u+v)=cu+cv Distributive property 8. (c+d)u=cu+du Distributive property 9. c(du)=(cd)u Associative property of multiplication 10. 1(u)=u Multiplicative identity property

At least one of the answers above is NOT correct. Let V(7,∞). For u, v € V and a ЄR define vector addition by uv := uv - 7(u+v) +56 and scalar multiplication by au := (u − 7)ª + 7. It can be shown that (V,, ) is a vector space over the scalar field R. Find the following: the sum: 8田9=9 the scalar multiple: -18 8 the additive inverse of 8: 日8 = 63 the zero vector: Ων 8 the additive inverse of x: 1 Bx +7 x-7

The figure shows a vector field F and three paths from P (-3,0) to Q= (3,0). The top and bottom paths T and B comprise a circle, and the middle path M is a line segment. Determine whether the following quantities are positive, negative, or zero, or answer true or false. Be sure you can explain your answers. (Click on graph to enlarge) (a) F dr is ? (b) F- dř is ? F. dr is ? (c) F-dr is 2 () (e) ? v True or False: F- dr () ? v True or False F is a gradient field.