If $\bar{a}, \bar{b}, \bar{c}$ are nonzero vectors along the coterminus edges of a parallelopiped with…

If $\bar{a}, \bar{b}, \bar{c}$ are nonzero vectors along the coterminus edges of a parallelopiped with volume 7 cubic units, then the volume of a parallelopiped with $\overline{\mathrm{a}}+\overline{\mathrm{b}}, \overline{\mathrm{b}}+\overline{\mathrm{c}}, \overline{\mathrm{c}}+\overline{\mathrm{a}}$ as the coterminus edges is
  1. 49 cubic units
  2. 2 cubic units
  3. 14 cubic units
  4. 7 cubic units

Solution

We have $[\bar{a} \cdot(\bar{b} \times \bar{c})]=7$ $[\bar{a}+\bar{b} \quad \bar{b}+\bar{c} \quad \bar{c}+\bar{a}]$ $=(\bar{a}+\bar{b}) \cdot[(\bar{b}+\bar{c}) \times(\bar{c}+\bar{a})]$ $=(\bar{a}+\bar{b}) \cdot[(\bar{b} \times \bar{c})+(\bar{b} \times \bar{a})+(\bar{c} \times \bar{c})+(\bar{c} \times \bar{a})]$ $=[\bar{a} \cdot(\bar{b} \times \bar{c})]+[\bar{a} \cdot(\bar{b} \times \bar{a})]+[\bar{a}(\bar{c} \times \bar{a})]+[\bar{b} \cdot(\bar{b} \times \bar{c})]+[\bar{b} \cdot(\bar{b} \times \bar{a})]+[\bar{b} \cdot(\bar{c} \times \bar{a})]$ $=[\bar{a} \cdot(\bar{b} \times \bar{c})]+0+[\bar{b} \cdot(\bar{c} \times \bar{a})]$ $=[\bar{a} \cdot(\bar{b} \times \bar{c})]+[\bar{a} \cdot(\bar{b} \times \bar{c})]=2[\bar{a} \cdot(\bar{b} \times \bar{c})]$ $=2(7)=14$

Asked in: MHT CET 2020 (19 Oct Shift 1)

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