Question

let (p=(0,0,0)), (q=(1, - 1,-2)), and (r=(-1,1,1)). find the area of the triangle (pqr). area =
let (t=(-5,-9,-10)), (u=(-2,9,-4)), and (v=(4,8,10)). find the area of the triangle (tuv). area = (\frac{sqrt{78921}}{2})

Explanation:

Step1: Find vectors $\overrightarrow{PQ}$ and $\overrightarrow{PR}$

$\overrightarrow{PQ}=(1 - 0,-1 - 0,-2 - 0)=(1,-1,-2)$
$\overrightarrow{PR}=(-1 - 0,1 - 0,1 - 0)=(-1,1,1)$

Step2: Calculate the cross - product $\overrightarrow{PQ}\times\overrightarrow{PR}$

$\overrightarrow{PQ}\times\overrightarrow{PR}=

$$\begin{vmatrix} \vec{i}&\vec{j}&\vec{k}\\ 1&-1&-2\\ -1&1&1 \end{vmatrix}$$

=\vec{i}(-1\times1 - (-2)\times1)-\vec{j}(1\times1-(-2)\times(-1))+\vec{k}(1\times1 - (-1)\times(-1))=\vec{i}( - 1 + 2)-\vec{j}(1 - 2)+\vec{k}(1 - 1)=\vec{i}+\vec{j}+0\vec{k}=(1,1,0)$

Step3: Find the magnitude of the cross - product

$|\overrightarrow{PQ}\times\overrightarrow{PR}|=\sqrt{1^{2}+1^{2}+0^{2}}=\sqrt{2}$

Step4: Calculate the area of triangle $PQR$

The area of triangle $PQR$ is $A=\frac{1}{2}|\overrightarrow{PQ}\times\overrightarrow{PR}|=\frac{\sqrt{2}}{2}$

Answer:

$\frac{\sqrt{2}}{2}$