Search any question & find its solution
Question:
Answered & Verified by Expert
If $\bar{a}=2 i+3 j-\hat{k}, \bar{b}=-i+2 j-4 \hat{k}$ and $\bar{c}=i+j+\hat{k}$, then $(\bar{a} \times \bar{b}) \cdot(\bar{a} \times \bar{c})=$
Options:
Solution:
2556 Upvotes
Verified Answer
The correct answer is:
$-74$
$\bar{a} \times \bar{b}=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -1 \\ -1 & 2 & -4\end{array}\right|=\hat{i}(-12+2)-\hat{j}(-8-1)+\hat{k}(4+3)=-10 \hat{i}+9 \hat{j}+7 \hat{k}$
$\begin{aligned} \bar{a} \times \bar{c} &=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -1 \\ 1 & 1 & 1\end{array}\right|=\hat{i}(3+1)-\hat{j}(2+1)+\hat{k}(2-3)=4 \hat{i}-3 \hat{j}-\hat{k} \\ \text { Here } \quad(\bar{a} \times \bar{b}) \cdot(\bar{a} \times \bar{c}) &=(-10 \hat{i}+9 \hat{j}+7 \hat{k}) \cdot(4 \hat{i}-3 \hat{j}-\hat{k}) \\ &=(-10)(4)+9(-3)+7(-1) \\ &=-40-27-7=-74 \end{aligned}$
$\begin{aligned} \bar{a} \times \bar{c} &=\left|\begin{array}{ccc}\hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & -1 \\ 1 & 1 & 1\end{array}\right|=\hat{i}(3+1)-\hat{j}(2+1)+\hat{k}(2-3)=4 \hat{i}-3 \hat{j}-\hat{k} \\ \text { Here } \quad(\bar{a} \times \bar{b}) \cdot(\bar{a} \times \bar{c}) &=(-10 \hat{i}+9 \hat{j}+7 \hat{k}) \cdot(4 \hat{i}-3 \hat{j}-\hat{k}) \\ &=(-10)(4)+9(-3)+7(-1) \\ &=-40-27-7=-74 \end{aligned}$
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.