벡터의 크기

$\left( 2 {\vec{a}} + {\vec{b}} \right) \bullet \left( 2 {\vec{a}} + {\vec{b}} \right)$$= 4 \left| {\vec{a}} \right| ^{2} + 4 {\vec{a}} \bullet {\vec{b}} + \left| {\vec{b}} \right| ^{2}$ $= 8 + 4 {\vec{a}} \bullet {\vec{b}} + \left| {\vec{b}} \right| ^{2} = 13$ $\cdots$$\cdots$ ㉠ $\left( {\vec{a}} - {\vec{b}} \right) \bullet \left( {\vec{a}} - {\vec{b}} \right)$$= \left| {\vec{a}} \right| ^{2} - 2 {\vec{a}} \bullet {\vec{b}} + \left| {\vec{b}} \right| ^{2}$ $= 2 - 2 {\vec{a}} \bullet {\vec{b}} + \left| {\vec{b}} \right| ^{2} = 1$ $\cdots$$\cdots$ ㉡ ㉠, ㉡에서 ${\vec{a}} \bullet {\vec{b}} = 1$, $\left| {\vec{b}} \right| = 1$ $\left( {\vec{a}} + {\vec{b}} \right) \bullet \left( {\vec{a}} + {\vec{b}} \right)$$= \left| {\vec{a}} \right| ^{2} + 2 {\vec{a}} \bullet {\vec{b}} + \left| {\vec{b}} \right| ^{2}$ $\displaystyle = \left( \sqrt{2} \right) ^{2} + 2 \times 1 + 1 ^{2} = 5$ 따라서 $\displaystyle \left| {\vec{a}} + {\vec{b}} \right| = \sqrt{5}$