Consider the vector field f(x, y)=(x^2+y^2)i+4xyj. compute the line integrals ∫c1 f⋅dr and ∫c2 f⋅dr, where c1 is parameterized by r(t)=ti+t^2j and c2 is parameterized by r(t)=ti+tj for 0 ≤ t ≤ 1. can you decide from your answers whether or not f is a gradient (conservative) vector field? ∫c1f⋅dr= ∫c2f⋅dr=

the value of v.w= -12.

step-by-step explanation:

we are given three vector r,v and w as:

r = < 5, -5, -2> ; v = < 2, -8, -8> ; w = < -2, 6, -5>

we know that r,v and w could also be represented as:

r=5i-5j-2k

v=2i-8j-8k

and w= -2i+6j-5k

where i,j and k are the unit vectors of x,y and z-axis respectively.

also we know that it has the following property:

i.i=1 ; j.j=1; k.k=1

and i.=j.i=i.k=k.i=j.k=k.j=0

i.e. when we multiply two vectors or we find it's dot product we need to only multiply the components of x coordinates with each other; y with each other and z with each other.

hence, v.w= (2i-8j- -2i+6j-5k)

                  = 2×(-2)+(-8)×(6)+(-8)×(-5)

                  = -4-48+40

                  = -12

hence, v.w= -12

correct answers below: