Micayla wants the print shop to reduce the size of her painting but keep the ratio of length to width so that it will fit her frame. Study the scale drawings to determine the proportional relationship between her painting and the frame she wants to use.

The width of the frame must be 12 inches (15in by 12in)

Step-by-step explanation:

Given

See attachment for painting and frame

Required

Determine the dimension of the frame

To do this, we make use of the following scale ratio.

For the painting,

For the frame,

Since she wants to maintain the same ratio; both scale must be the same.

i.e.

Express as fraction

Make w the subject

The width of the frame must be 12 inches i.e. 15in by 12in frame

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