Kieran, Jermaine and Chris play football. Kieran has scored 8 more goals than Chris.
Jermaine has scored 5 more goals than Kieran.
Altogether they have scored 72 goals.
How many goals did they each score?

Step-by-step explanation:

Find an equation that explains the problem. We can see how many goals the players have scored in comparison to each other. The person with the fewest goals is Chris. Let us say he score x goals. Then we can see that Kieran has scored 8 more than Chris, i.e. x + 8. Jermaine scored 5 more goals than Kieran, therefore x + 8 + 5 = x + 13. Now we have the amount of goals each person scored relative to each other. The total number of goals scored was 72, thus all of these amounts must equal 72.  

x + (x + 8) + (x + 13) = 72

Now we can rearrange and solve for x.

3x + 21 = 72

3x = 51

x = 17

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

step-by-step explanation: