Suppose that u,vu,v and w are vectors such that ⟨u,v⟩=6, ⟨v,w⟩=−7, ⟨u,w⟩=13⟨ ∣∣u∣∣=1,∣∣v∣∣=6,∣∣w∣∣=19∣∣u∣∣=1, given expression ⟨u+v,u+w⟩.

Suppose that u,vu,v and w are vectors such that $$⟨u,v⟩=6, ⟨v,w⟩=−7, ⟨u,w⟩=13∣∣u∣∣=1,∣∣v∣∣=6,∣∣w∣∣=19∣∣u∣∣=1$$, given expression ⟨u+v,u+w⟩.

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Velsenw

Given that u,v and w are vectors such that $$\displaystyle{\left({u},{v}\right)}={6},{\left({v},{w}\right)}=-{7},{\left({u},{w}\right)}={13},{\left|{\left|{u}\right|}\right|}={1},{\left|{\left|{v}\right|}\right|}={6},{\left|{\left|{w}\right|}\right|}={19}$$
$$(u+v,v+w)=(u,v)+(u,w)+(v,v)+(v,w) =6+13+(6^2)-7$$

$$=12+36 =48.$$
Again $$\displaystyle{\left({u}+{v},{u}+{w}\right)}={\left({u},{u}\right)}+{\left({u},{w}\right)}+{\left({v},{u}\right)}+{\left({v},{w}\right)}$$ =1+13+6-7 =13.