Question

Calculate the area of a triangle whose vertices are given by (3,-1,2),(1,-1,2),(4,-2,1)

Answer 1

Let

$$A=(3,-1,2),\quad B=(1,-1,2), \quad C=(4,-2,1)$$

So

$$\overrightarrow{AB}=(1-3,-1-(-1),2-2)=(-2,0,0)\\ \overrightarrow{AC}=(4-3,-2-(-1),1-2)=(1,-1,-1)$$

The vector product

$$[\overrightarrow{AB}\times \overrightarrow{AC}]=\begin{vmatrix} \hat i & \hat j & \hat k\\ -2 & 0 & 0\\ 1 & -1 & -1 \end{vmatrix}=-2\hat j+2\hat k$$

The area of the triangle

$$A=\frac{1}{2}|[\overrightarrow{AB}\times \overrightarrow{AC}]|=\frac{1}{2}\sqrt{(-2)^2+2^2}=\sqrt{2}$$