If the sum of the squares of nonnegative real numbers a, b, and c is 39, and ab + bc + ca = 21, then what is the sum of a, b, and c?

Step-by-step explanation:

a² + b² + c² = 39

ab + bc + ca = 21

(a + b + c)² = a² + b² +c² + 2[ab + bc +ca]

(a + b + c)² = 39 + 2*21

                 = 39 + 42

(a+ b + c)² = 81

a +b + c = √81 = √9*9

a + b + c = 9

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