3) The points (2,2,0), (-2,4,1) and (-1,7,3) are the vertices of a triangle.
a) Determine the area of the triangle.
b) Determine the internal angles of the triangle.
c) Determine the length of each side of the triangle.
d) Represent the triangle on the space

We will break down this problem step by step. Given the three points \( A(2, 2, 0) \), \( B(-2, 4, 1) \), and \( C(-1, 7, 3) \), we can determine the following:

### a) Area of the Triangle

The area of a triangle formed by three points in 3D space can be calculated using the cross product of two vectors from the points, followed by calculating half the magnitude of the resulting vector.

The vectors \( \vec{AB} \) and \( \vec{AC} \) are:
\[
\vec{AB} = B - A = (-2 - 2, 4 - 2, 1 - 0) = (-4, 2, 1)
\]
\[
\vec{AC} = C - A = (-1 - 2, 7 - 2, 3 - 0) = (-3, 5, 3)
\]

The cross product \( \vec{AB} \times \vec{AC} \) gives a vector orthogonal to the triangle's plane.

\[
\vec{AB} \times \vec{AC} = \begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
-4 & 2 & 1 \\
-3 & 5 & 3 \\
\end{vmatrix}
= \hat{i}(2 \times 3 - 1 \times 5) - \hat{j}(-4 \times 3 - 1 \times (-3)) + \hat{k}(-4 \times 5 - 2 \times (-3))
\]

I'll calculate this next.The cross product \( \vec{AB} \times \vec{AC} = (1, 9, -14) \), and the magnitude of this vector is approximately \( 16.67 \). Thus, the area of the triangle is:

\[
\text{Area} = \frac{1}{2} \times 16.67 \approx 8.34
\]

### b) Internal Angles of the Triangle

To calculate the internal angles, we first need the lengths of the sides of the triangle.

### c) Length of Each Side of the Triangle

The lengths of the sides can be calculated using the distance formula between two points \( A(x_1, y_1, z_1) \) and \( B(x_2, y_2, z_2) \):

\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}
\]

Let me calculate the lengths of sides \( AB \), \( BC \), and \( AC \).It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to explain the steps to manually calculate the lengths of the sides and proceed from there, or do you want to try another question?

Here are 5 questions to explore based on this problem:
1. How is the cross product used to determine the area of a triangle in 3D?
2. Can you manually calculate the side lengths of the triangle using the distance formula?
3. What is the significance of the cross product's direction in determining a triangle's properties?
4. How do you find the angles of a triangle in 3D using side lengths?
5. How does the determinant of a matrix relate to the cross product in vector operations?

**Tip:** To calculate the angles of a triangle from its side lengths, you can apply the **cosine rule**.

The points (2,2,0), (-2,4,1) and (-1,7,3) are the vertices of a triangle.
 a) Determine the area of the triangle.
 b) Determine the internal angles of the triangle.
 c) Determine the length of each side of the triangle.
 d) Represent the triangle in space.

Learn how to calculate the area, internal angles, and side lengths of a triangle in 3D space formed by the points (2,2,0), (-2,4,1), and (-1,7,3). This problem covers vector operations, cross product, distance formula, and cosine rule. Suitable for students in Grades 10-12.

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