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  1. 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:



    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

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