Billy wants to make trail mix using almonds, peanuts, and raisins. he wants to mix one part almonds, two parts peanuts, and three parts raisins. almonds cost $12 per pound, peanuts cost $8 per pound, and raisins cost $6 per pound. let x represent the number of pounds of almonds.
a. billy has $23 to spend on the trail mix. write an equation to find the number of pounds of almonds billy will be able to buy.
b. determine how many pounds of trail mix he can make.
show your work.

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