Question

The area of the triangle formed by $2\hat{\mathrm{ı}}+\hat{\mathrm{ȷ}}-\hat{\mathrm{k}}$ and $\hat{\mathrm{ȷ}}+\hat{\mathrm{ȷ}}+\hat{\mathrm{k}}$ is

• A. $3\mathrm{sq}.\mathrm{units}$
• B. $2\sqrt{3}\mathrm{sq}.\mathrm{units}$
• C. $2\sqrt{14}\mathrm{sq}.\mathrm{units}$
• D. $\frac{\sqrt{14}}{2}\mathrm{sq}.\mathrm{units}$

Answer 1

The correct option is D

$\frac{\sqrt{14}}{2}\mathrm{sq}.\mathrm{units}$

Explanation for correct option:

Finding the area of the triangle

Given,

$$\begin{gathered} \overrightarrow{\mathrm{A}}=2\hat{\mathrm{ı}}+\hat{\mathrm{ȷ}}-\hat{\mathrm{k}} \\ \overrightarrow{\mathrm{B}}=\hat{\mathrm{ȷ}}+\hat{\mathrm{ȷ}}+\hat{\mathrm{k}} \end{gathered}$$

Calculating the area of the triangle:

The area of the triangle is,

$$\begin{gathered} =\frac{1}{2}|\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}}| \\ =\frac{1}{2}\left|\begin{array}{c}\left|\begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ 2& 1& -1\\ 1& 1& 1\end{array}\right|\end{array}\right| \\ =\frac{1}{2}\left[\hat{\mathrm{ȷ}}\right(1+1)-\hat{\mathrm{ȷ}}(2+1)+\hat{\mathrm{k}}(2-1\left)\right] \\ =\frac{1}{2}|2\hat{\mathrm{ı}}-3\hat{\mathrm{ȷ}}+\hat{\mathrm{k}}| \\ =\frac{1}{2}\sqrt{2^2+(-3{)}^2+1^2} \\ =\frac{\sqrt{14}}{2}\text{sq.units} \end{gathered}$$

Hence, the correct answer is option (D).