Questions: Determine the area of a triangle with points A(0, 1, 6), B(6, 2, -5), and C(1, -6, 6). Area = 7899 / 2 ×

Solution Steps

Let the points be defined as follows: \[ A = (0, 1, 6), \quad B = (6, 2, -5), \quad C = (1, -6, 6) \]

Calculate the vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\): \[ \overrightarrow{AB} = B - A = (6 - 0, 2 - 1, -5 - 6) = (6, 1, -11) \] \[ \overrightarrow{AC} = C - A = (1 - 0, -6 - 1, 6 - 6) = (1, -7, 0) \]

Calculate the cross product \(\overrightarrow{AB} \times \overrightarrow{AC}\): \[ \overrightarrow{AB} \times \overrightarrow{AC} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 6 & 1 & -11 \\ 1 & -7 & 0 \end{vmatrix} = (-77, -11, -43) \]

Calculate the magnitude of the cross product: \[ \text{Magnitude} = \sqrt{(-77)^2 + (-11)^2 + (-43)^2} = \sqrt{5929 + 121 + 1849} = \sqrt{5889} \approx 88.87631855561975 \]

The area \(A\) of the triangle is given by: \[ A = \frac{1}{2} \times \text{Magnitude} = \frac{1}{2} \times 88.87631855561975 \approx 44.43815927780987 \]

Final Answer