Answer
$-24$
Work Step by Step
Let $u=\left\langle a,b\right \rangle$ and $v=\left\langle c,d\right \rangle$ be two vectors, then their dot product is given by
$u\cdot v=\left\langle a,b\right \rangle \cdot \left\langle c,d\right \rangle=ac+bd$
and subtraction by $u-v=\left\langle a,b\right \rangle-\left\langle c,d\right \rangle=\left\langle a-c,b-d\right \rangle$
Now for $u=\left\langle -2,1\right \rangle$ , $v=\left\langle 3,4\right \rangle$ and $w=\left\langle -5,12\right \rangle$ we have
$v-w=\left\langle 3,4\right \rangle -\left\langle -5,12\right \rangle$
$v-w= \left\langle 3-(-5),4-12\right \rangle=\left\langle 8,-8\right \rangle$
and
$u\cdot (v-w)=\left\langle -2,1\right \rangle \cdot \left\langle 8,-8\right \rangle$
$u\cdot (v-w)=(-2)(8)+(1)(-8)=-16-8=-24$ .
Therefore, $u\cdot (v-w)=-24$ .
