Using vector method, find the area of the triangle whose vertices $(1, 1, 2), (2, 3, 5)$ and $(1, 5, 5)$ .

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Solution

Let, $\to O A = i + j + 2 k \to O B = 2 i + 3 j + 5 k \to O C = i + 5 j + 5 k$
$\to A B = \to O B - \to O A = (2 i + 3 j + 5 k) - (i + j + 2 k) = i + 2 j + 3 k$
$\to B C = \to O C - \to O B = (i + 5 j + 5 k) - (2 i + 3 j + 5 k) = - i + 2 j$
Now, Area of the Triangle $= \frac{1}{2} | \to A B$ x $\to B C |$
$= \frac{1}{2} ∣ ∣ ∣ ∣ \begin{matrix} i & j & k 1 & 2 & 3 - 1 & 2 & 0 \end{matrix} ∣ ∣ ∣ ∣ = \frac{1}{2} | - 6 i - 3 j + 4 k | = \frac{1}{2} | \sqrt{36 + 9 + 16} | = \frac{\sqrt{61}}{2}$

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