The text in the image provides the coordinates of three points in a triangle, labeled A, B, and C:

• $A(0; 2; 1)$
• $B(-3; 2; 4)$
• $C(-1; 0; 2)$

The task is to find the cosine of the angle at vertex $C$ in the triangle $ABC$.

To solve this, we use the formula for the cosine of the angle between two vectors in 3D space. The cosine of the angle between two vectors $\mathbf{v_1}$ and $\mathbf{v_2}$ is given by the formula:

$\cos \theta = \frac{\mathbf{v_1} \cdot \mathbf{v_2}}{\|\mathbf{v_1}\| \|\mathbf{v_2}\|}$

Where:

• $\mathbf{v_1}$ and $\mathbf{v_2}$ are the vectors representing the sides of the triangle from vertex $C$,
• $\mathbf{v_1} \cdot \mathbf{v_2}$ is the dot product of the two vectors, and
• $\|\mathbf{v_1}\|$ and $\|\mathbf{v_2}\|$ are the magnitudes (lengths) of the vectors.

First, we need to find the vectors representing the sides $\overrightarrow{BC}$ and $\overrightarrow{AC}$.

Step 1: Calculate the vectors $\overrightarrow{BC}$ and $\overrightarrow{AC}$

$\overrightarrow{BC} = C - B = (-1 - (-3), 0 - 2, 2 - 4) = (2, -2, -2)$ $\overrightarrow{AC} = C - A = (-1 - 0, 0 - 2, 2 - 1) = (-1, -2, 1)$

Step 2: Calculate the dot product $\overrightarrow{BC} \cdot \overrightarrow{AC}$

$\overrightarrow{BC} \cdot \overrightarrow{AC} = (2)(-1) + (-2)(-2) + (-2)(1) = -2 + 4 - 2 = 0$

Step 3: Find the magnitudes of the vectors

$\|\overrightarrow{BC}\| = \sqrt{(2)^2 + (-2)^2 + (-2)^2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3}$ $\|\overrightarrow{AC}\| = \sqrt{(-1)^2 + (-2)^2 + (1)^2} = \sqrt{1 + 4 + 1} = \sqrt{6}$

Step 4: Calculate the cosine of the angle $\theta$

Since the dot product is 0, this means the vectors $\overrightarrow{BC}$ and $\overrightarrow{AC}$ are orthogonal, and hence the cosine of the angle between them is:

$\cos \theta = \frac{0}{(2\sqrt{3})(\sqrt{6})} = 0$

Thus, the cosine of the angle at vertex $C$ in triangle $ABC$ is 0, indicating that the angle at vertex $C$ is 90 degrees.

Would you like more details on any part of this solution or further questions? Here are some related questions for you:

1. How do you calculate the dot product of two vectors?
2. What are the properties of orthogonal vectors?
3. How do you compute the magnitude of a vector in 3D space?
4. What other methods can you use to find angles in triangles in 3D?
5. How do you determine if two vectors are parallel or orthogonal in general?

Tip: When calculating dot products, remember that the result gives insight into the angle between the vectors, with a dot product of zero indicating perpendicular vectors.