202601291

6.其他

$已知非零向量 \vec{a}, \vec{b}, \vec{c} 满足 3 \vec{a} + 4 \vec{b} + 5 \vec{c} = \vec{0}, | \vec{a} | = | \vec{b} | = | \vec{c} |, 求 \vec{a} \cdot \vec{b}$

$3 \vec{a} + 4 \vec{b} = - 5 \vec{c}$
$(3 \vec{a} + 4 \vec{b})^{2} = 25 {\vec{c}}^{2}$
$9 {\vec{a}}^{2} + 24 \vec{a} \cdot \vec{b} + 16 \vec{b^{2}} = 25 {\vec{c}}^{2}$
$| \vec{a} | = | \vec{b} | = | \vec{c} |$
$∴ {\vec{a}}^{2} = {\vec{b}}^{2} = {\vec{c}}^{2}$
$∴ 9 + 24 \vec{a} \cdot \vec{b} + 16 = 25$
$\vec{a} \cdot \vec{b} = 0$