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6. 已知向量 $\boldsymbol{a}, \boldsymbol{b}$ 满足 $|a|=5,|b|=6, a \cdot b=-6$, 则 $\cos \langle\boldsymbol{a}, \boldsymbol{a}+\boldsymbol{b}\rangle=(\quad)$

Gaokao · Math · previous-year question

  1. A.$-\frac{31}{35}$
  2. B.$-\frac{19}{35}$
  3. C.$\frac{17}{35}$
  4. D.$\frac{19}{35}$correct

Answer

D. $\frac{19}{35}$

Explanation

【详解】 $\because|\vec{a}|=5,|\vec{b}|=6, \vec{a} \cdot \vec{b}=-6, \quad \therefore \vec{a} \cdot(\vec{a}+\vec{b})=|\vec{a}|^{2}+\vec{a} \cdot \vec{b}=5^{2}-6=19$ ． $|\vec{a}+\vec{b}|=\sqrt{(\vec{a}+\vec{b})^{2}}=\sqrt{\vec{a}^{2}+2 \vec{a} \cdot \vec{b}+\vec{b}^{2}}=\sqrt{25-2 \times 6+36}=7$ 因此, $\cos \langle\vec{a}, \vec{a}+\vec{b}\rangle=\frac{\vec{a} \cdot(\vec{a}+\vec{b})}{|\vec{a}| \cdot|\vec{a}+\vec{b}|}=\frac{19}{5 \times 7}=\frac{19}{35}$. 故选: D.

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