Question

Using vectors, find the area of the triangle with vertices A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5)

Using vectors, find the area of the triangle with vertices A(1, 2, 3), B(2, -1, 4) and C(4, 5, -1)

Answer 1

The given points are : A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5)

Let O be the origin.

$$\begin{gathered} \mathrm{Position}\mathrm{vector}\mathrm{of}\mathrm{A}=\overrightarrow{\mathrm{OA}}=\hat{\mathrm{i}}+\hat{\mathrm{j}}+2\hat{\mathrm{k}} \\ \mathrm{position}\mathrm{vector}\mathrm{of}\mathrm{B}=\overrightarrow{\mathrm{OB}}=2\hat{\mathrm{i}}+3\hat{\mathrm{j}}+5\hat{\mathrm{k}} \\ \mathrm{position}\mathrm{vector}\mathrm{of}\mathrm{C}=\overrightarrow{\mathrm{OC}}=\hat{\mathrm{i}}+5\hat{\mathrm{j}}+5\hat{\mathrm{k}} \\ \overrightarrow{\mathrm{AB}}=\overrightarrow{\mathrm{OB}}-\overrightarrow{\mathrm{OA}}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}} \\ \overrightarrow{\mathrm{AC}}=\overrightarrow{\mathrm{OC}}-\overrightarrow{\mathrm{OA}}=4\hat{\mathrm{j}}+3\hat{\mathrm{k}} \end{gathered}$$



$$\begin{gathered} \mathrm{Area}=\frac{1}{2}\left|\overrightarrow{\mathrm{AB}}\mathrm{X}\overrightarrow{\mathrm{AC}}\right| \\ \mathrm{Now},\overrightarrow{\mathrm{AB}}\mathrm{X}\overrightarrow{\mathrm{AC}}=\left|\begin{array}{ccc}\hat{\mathrm{i}}& \hat{\mathrm{j}}& \hat{\mathrm{k}}\\ 1& 2& 3\\ 0& 4& 3\end{array}\right|=-6\hat{\mathrm{i}}-3\hat{\mathrm{j}}+4\hat{\mathrm{k}} \\ \mathrm{Area}=\frac{1}{2}|-6\hat{\mathrm{i}}-3\hat{\mathrm{j}}+4\hat{\mathrm{k}}|=\frac{1}{2}\sqrt{36+9+16}=\frac{\sqrt{61}}{2}\mathrm{sq}\mathrm{units} \end{gathered}$$