Vectors 3 Question 12
12. If $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ are three non-coplanar vectors, then $(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \cdot[(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \times(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{c}})]$ equals
(1995, 2M)
(a) 0
(b) $[\overrightarrow{\mathbf{c}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]$
(c) $2 \cdot[\overrightarrow{\mathbf{c}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]$
(d) $[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]$
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Answer:
Correct Answer: 12. (d)
Solution:
- $(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \cdot[(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}) \times(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{c}})]$
$=(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}}) \cdot[\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}]$
$={\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}})+\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})+\overrightarrow{\mathbf{a}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})}+${$\overrightarrow{\mathbf{b}} \cdot \overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}}$
$+\overrightarrow{\mathbf{b}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})+\overrightarrow{\mathbf{b}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})+{\overrightarrow{\mathbf{c}} \cdot(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}})+\overrightarrow{\mathbf{c}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})+\overrightarrow{\mathbf{c}} \cdot(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})}$
$=[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]+[\overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{c}}]+[\overrightarrow{\mathbf{c}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{a}}]=[\overrightarrow{\mathbf{a}} \overrightarrow{\mathbf{b}} \overrightarrow{\mathbf{c}}]$
