Hannah examined these multiples of 9: 18, 45, 63, 27, 81, 108, 216. she claimed that the sum of the digits in any multiple of 9 will add to 9. do you agree or disagree? justify your decision.

Yes, the sum of the digits in any multiple of 9 will add to 9 always.

Explanation

If we multiply any number by 9, then the result is said to be "multiple of 9".

So, lets take any number like 1456 and now we will multiply it by 9.

Thus,  

That means 13104 is a multiple of 9.

Now the sum of the digits of this 13104 will be:  

So, we can say that the sum of the digits in any multiple of 9 will add to 9 always.

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