Vectors 3 Question 23
23. If $\overrightarrow{\mathbf{A}}, \overrightarrow{\mathbf{B}}, \overrightarrow{\mathbf{C}}$ are three non-coplanar vectors, then $\frac{\overrightarrow{\mathbf{A}} \cdot(\overrightarrow{\mathbf{B}} \times \overrightarrow{\mathbf{C}})}{(\overrightarrow{\mathbf{C}} \times \overrightarrow{\mathbf{A}}) \cdot \overrightarrow{\mathbf{B}}}+\frac{\overrightarrow{\mathbf{B}} \cdot(\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{C}})}{\overrightarrow{\mathbf{C}} \cdot(\overrightarrow{\mathbf{A}} \times \overrightarrow{\mathbf{B}})}=\ldots$
(1985, 2M)
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Answer:
Correct Answer: 23. (0)
Solution:
- $\frac{\overrightarrow{A} \cdot(\overrightarrow{\mathbf{B}} \times \overrightarrow{\mathbf{C}})}{(\overrightarrow{\mathbf{C}} \times \overrightarrow{\mathbf{A}}) \cdot \overrightarrow{\mathbf{B}}}+\frac{\overrightarrow{B} \cdot(\overrightarrow{A} \times \overrightarrow{\mathbf{C}})}{\overrightarrow{C} \cdot(\overrightarrow{A} \times \overrightarrow{\mathbf{B}})}$
$$ =\frac{[\overrightarrow{A} \overrightarrow{B} \overrightarrow{C}]}{[\overrightarrow{C} \overrightarrow{A} \overrightarrow{B}]}+\frac{[\overrightarrow{B} \overrightarrow{A} \overrightarrow{C}]}{[\overrightarrow{C} \overrightarrow{A} \overrightarrow{B}]}=\frac{[\overrightarrow{A} \overrightarrow{B} \overrightarrow{C}]-[\overrightarrow{A} \overrightarrow{B} \overrightarrow{C}]}{[\overrightarrow{C} \overrightarrow{A} \overrightarrow{B}]}=0 $$
