Question

Calculate the area of the triangle determined by the two vectors $\overrightarrow{A}=\hat{i}-2\hat{j}+\hat{k}$ and $\overrightarrow{B}=2\hat{i}+3\hat{j}-\hat{k}$

• A. $\hat{i}+3\hat{j}-7\hat{k}$
• B. $\frac{1}{2}(-\hat{i}-3\hat{j}+7\hat{k})$
• C. $\frac{1}{2}(-\hat{i}+3\hat{j}+7\hat{k})$
• D. $-\hat{i}+3\hat{j}+7\hat{k}$

Answer 1

The correct option is C $\frac{1}{2}(-\hat{i}+3\hat{j}+7\hat{k})$

We know that the area of the triangle is given by $\frac{1}{2}(\overrightarrow{A}\times \overrightarrow{B})$
Thus, we have $\overrightarrow{A}\times \overrightarrow{B}$ as
$\left[\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -2& 1\\ 2& 3& -1\end{array}\right]$
$\Rightarrow (2-3)\hat{i}-(-1-2)\hat{j}+(3-(-4\left)\right)\hat{k}$
$\Rightarrow -\hat{i}+3\hat{j}+7\hat{k}$
Hence, the area of the triangle is
$\frac{1}{2}(\overrightarrow{A}\times \overrightarrow{B})=\frac{1}{2}(-\hat{i}+3\hat{j}+7\hat{k})$