Search any question & find its solution
Question:
Answered & Verified by Expert
If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}+\hat{j}-\hat{k}$, then what is
$\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times(\vec{c}+\vec{a})+\vec{c} \times(\vec{a}+\vec{b})$ equal to?
Options:
$\vec{a} \times(\vec{b}+\vec{c})+\vec{b} \times(\vec{c}+\vec{a})+\vec{c} \times(\vec{a}+\vec{b})$ equal to?
Solution:
2072 Upvotes
Verified Answer
The correct answer is:
$\overrightarrow{0}$
Since, $\overrightarrow{\mathbf{a}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \overrightarrow{\mathbf{b}}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\overrightarrow{\mathbf{c}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$
$\Rightarrow \overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}})+\overrightarrow{\mathbf{b}} \times(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}})+\overrightarrow{\mathbf{c}} \times(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}})$
$=(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})+(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}})+(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})+(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})+(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})+(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{b}})$
$=(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})-(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})+(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})-(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})+(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})-(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})=0$
$\Rightarrow \overrightarrow{\mathbf{a}} \times(\overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{c}})+\overrightarrow{\mathbf{b}} \times(\overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{a}})+\overrightarrow{\mathbf{c}} \times(\overrightarrow{\mathbf{a}}+\overrightarrow{\mathbf{b}})$
$=(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})+(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{c}})+(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})+(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{a}})+(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})+(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{b}})$
$=(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})-(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})+(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})-(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}})+(\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}})-(\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}})=0$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.