题目
2.设a,b,c为单位向量，且满足a+b+c=0，求a·b+b·c+c·a.3.已知M_(1)(1,-1,2),M_(2)(3,3,1)和M_(3)(3,1,3).求与overrightarrow(M_{1)M_(2)},overrightarrow(M_{2)M_(3)}同时垂直的单位向量.
2.设a,b,c为单位向量，且满足a+b+c=0，求a·b+b·c+c·a.
3.已知$M_{1}(1,-1,2)$,$M_{2}(3,3,1)$和$M_{3}(3,1,3)$.求与$\overrightarrow{M_{1}M_{2}}$,$\overrightarrow{M_{2}M_{3}}$同时垂直的单位向量.
题目解答
答案
**问题2：**
已知 $\mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0}$，两边平方得：
\[
(\mathbf{a} + \mathbf{b} + \mathbf{c})^2 = 0 \implies \mathbf{a}^2 + \mathbf{b}^2 + \mathbf{c}^2 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}) = 0
\]
由单位向量性质 $\mathbf{a}^2 = \mathbf{b}^2 = \mathbf{c}^2 = 1$，代入得：
\[
3 + 2(\mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a}) = 0 \implies \mathbf{a} \cdot \mathbf{b} + \mathbf{b} \cdot \mathbf{c} + \mathbf{c} \cdot \mathbf{a} = -\frac{3}{2}
\]
**答案：** $\boxed{-\frac{3}{2}}$
**问题3：**
计算向量：
\[
\overrightarrow{M_1M_2} = (2, 4, -1), \quad \overrightarrow{M_2M_3} = (0, -2, 2)
\]
求叉积：
\[
\overrightarrow{M_1M_2} \times \overrightarrow{M_2M_3} = (6, -4, -4)
\]
单位化：
\[
$ \overrightarrow{M_1M_2} \times \overrightarrow{M_2M_3} $ = 2\sqrt{17} \implies \pm \left( \frac{3}{\sqrt{17}}, -\frac{2}{\sqrt{17}}, -\frac{2}{\sqrt{17}} \right)
\]
**答案：** $\boxed{\pm \left( \frac{3}{\sqrt{17}}, -\frac{2}{\sqrt{17}}, -\frac{2}{\sqrt{17}} \right)}$