Question

1. find the area of the triangle with vertices a, b, and c : (a) a = (2, 3, 4), b = (4, 5, 6), c = (7, 7, 7). (b) a = (2, 3, 4), b = (6, 5, 4), c = (8, 8, 8).

Explanation:

Part (a)

Step1: Find vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$

$\overrightarrow{AB} = B - A = (4 - 2, 5 - 3, 6 - 4) = (2, 2, 2)$
$\overrightarrow{AC} = C - A = (7 - 2, 7 - 3, 7 - 4) = (5, 4, 3)$

Step2: Compute the cross product $\overrightarrow{AB} \times \overrightarrow{AC}$

$$ \overrightarrow{AB} \times \overrightarrow{AC} =
LATEXBLOCK0
= \mathbf{i}(2 \cdot 3 - 2 \cdot 4) - \mathbf{j}(2 \cdot 3 - 2 \cdot 5) + \mathbf{k}(2 \cdot 4 - 2 \cdot 5) $$

$$ = \mathbf{i}(6 - 8) - \mathbf{j}(6 - 10) + \mathbf{k}(8 - 10) = (-2, 4, -2) $$

Step3: Find the magnitude of the cross product

$$ \|\overrightarrow{AB} \times \overrightarrow{AC}\| = \sqrt{(-2)^2 + 4^2 + (-2)^2} = \sqrt{4 + 16 + 4} = \sqrt{24} = 2\sqrt{6} $$

Step4: Calculate the area of the triangle

The area is $\frac{1}{2} \|\overrightarrow{AB} \times \overrightarrow{AC}\| = \frac{1}{2} \cdot 2\sqrt{6} = \sqrt{6}$

Part (b)

Step1: Find vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$

$\overrightarrow{AB} = B - A = (6 - 2, 5 - 3, 4 - 4) = (4, 2, 0)$
$\overrightarrow{AC} = C - A = (8 - 2, 8 - 3, 8 - 4) = (6, 5, 4)$

Step2: Compute the cross product $\overrightarrow{AB} \times \overrightarrow{AC}$

$$ \overrightarrow{AB} \times \overrightarrow{AC} =
LATEXBLOCK0
= \mathbf{i}(2 \cdot 4 - 0 \cdot 5) - \mathbf{j}(4 \cdot 4 - 0 \cdot 6) + \mathbf{k}(4 \cdot 5 - 2 \cdot 6) $$

$$ = \mathbf{i}(8 - 0) - \mathbf{j}(16 - 0) + \mathbf{k}(20 - 12) = (8, -16, 8) $$

Step3: Find the magnitude of the cross product

$$ \|\overrightarrow{AB} \times \overrightarrow{AC}\| = \sqrt{8^2 + (-16)^2 + 8^2} = \sqrt{64 + 256 + 64} = \sqrt{384} = 8\sqrt{6} $$

Step4: Calculate the area of the triangle

The area is $\frac{1}{2} \|\overrightarrow{AB} \times \overrightarrow{AC}\| = \frac{1}{2} \cdot 8\sqrt{6} = 4\sqrt{6}$

Answer:

s:
(a) $\boldsymbol{\sqrt{6}}$
(b) $\boldsymbol{4\sqrt{6}}$