Vectors 3 Question 25
25. For any three vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$,
$$ (\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}) \cdot{(\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{c}}) \times(\overrightarrow{\mathbf{c}}-\overrightarrow{\mathbf{a}})}=2 \overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}) $$
$(1983,1 M)$
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Answer:
Correct Answer: 25. True
Solution:
- $(\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}) \cdot{(\overrightarrow{\mathbf{b}}-\overrightarrow{\mathbf{c}}) \times(\overrightarrow{\mathbf{c}}-\overrightarrow{\mathbf{a}})}=(\overrightarrow{\mathbf{a}}-\overrightarrow{\mathbf{b}}) \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}-\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})$
$$ \begin{aligned} =\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})-\overrightarrow{\mathbf{b}} \cdot(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}) & =[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]+[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}] \\ & =2[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]=2 \overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}) \end{aligned} $$
Hence, it is a true statement.
