Question

If $a=2\hat{i}+3\hat{j}-\hat{k},b=-\hat{i}+2j-4\hat{k}$and $c=\hat{i}+\hat{j}+\hat{k},$then what is the value of $(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{a}\times \overrightarrow{c})$$?$

• A. $47$
• B. $74$
• C. $-74$
• D. None of these

Answer 1

The correct option is C

$-74$

Step1. Find the value of $(\overrightarrow{a}\times \overrightarrow{b})$ and $(\overrightarrow{a}\times \overrightarrow{c})$:

Given that $a=2\hat{i}+3\hat{j}-\hat{k},b=-\hat{i}+2j-4\hat{k}$ and $c=\hat{i}+\hat{j}+\hat{k},$

Now, $(\overrightarrow{a}\times \overrightarrow{b})=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& -1\\ -1& 2& -4\end{array}\right|$

$$\begin{gathered} =\hat{i}(-12+2)-\widehat{j(}-8-1)+\hat{k}(4+3) \\ =-10\hat{i}+9\hat{j}+7\hat{k} \end{gathered}$$

Now, $(\overrightarrow{a}\times \overrightarrow{c})=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 2& 3& -1\\ 1& 1& 1\end{array}\right|$

$$\begin{gathered} =\hat{i}(3+1)-\widehat{j(}2+1)+\hat{k}(2-3) \\ =4\hat{i}-3\hat{j}-\hat{k} \end{gathered}$$

Step2. Find the value of $(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{a}\times \overrightarrow{c})$:

$(\overrightarrow{a}\times \overrightarrow{b}).(\overrightarrow{a}\times \overrightarrow{c})=(-10\hat{i}+9\hat{j}+7\hat{k}).(4\hat{i}-3\hat{j}-\hat{k})$

$$\begin{gathered} =-10\times 4+9\times -3+7\times -1 \\ =-40-27-7 \\ =-74 \end{gathered}$$

Hence, the correct option is C.