https://r2.erweima.ai/mathbot/image/1725938778040x7mzekub.jpg

To solve the problem, let's analyze the vectors in the triangle based on the image provided:

### Problem Analysis:
1. We have a right triangle with sides of lengths 3, 4, and 5.
2. The vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) correspond to the sides of this triangle.
3. The vectors sum to zero: \(\vec{A} + \vec{B} + \vec{C} = 0\).

### Vector Assignments:
- Let \(\vec{A}\) be the vector along the side of length 3, pointing upwards (from the right-angle vertex to the vertex labeled \(C\)).
- Let \(\vec{B}\) be the vector along the side of length 4, pointing to the left (horizontal side).
- Let \(\vec{C}\) be the hypotenuse vector of length 5.

### Calculation of Dot Products:
**(a) \(\vec{A} \cdot \vec{B}\)**:
- Since \(\vec{A}\) is perpendicular to \(\vec{B}\), \(\vec{A} \cdot \vec{B} = 0\).

**(b) \(\vec{A} \cdot \vec{C}\)**:
- Use the projection formula, where \(\vec{A} \cdot \vec{C} = |\vec{A}| |\vec{C}| \cos(\theta)\).
- Given \(\vec{A}\) and \(\vec{C}\) form the angle at point \(B\) (90 degrees), \(\theta = 0^\circ\), so \(\cos(\theta) = 0\), hence \(\vec{A} \cdot \vec{C} = 0\).

**(c) \(\vec{B} \cdot \vec{C}\)**:
- Since \(\vec{B}\) is perpendicular to \(\vec{C}\), \(\vec{B} \cdot \vec{C} = 0\).

All calculations show that the vectors are orthogonal, making all dot products zero.

Would you like further details on any of the steps or concepts? Here are some related questions for further exploration:

1. What are the properties of dot products in vector mathematics?
2. How do you determine the direction of vectors in coordinate systems?
3. Can you calculate vector magnitudes and directions from given coordinates?
4. How do you apply the Pythagorean theorem in vector problems?
5. What is the significance of vector orthogonality in physics and engineering?

**Tip:** When working with vectors, always verify if they are orthogonal, as this simplifies dot product calculations.

Tres vectores suman cero, como se muestra en el triángulo rectángulo de la figura. Calcular (a) A · B, (b) A · C, (c) B · C.

Learn how to calculate the dot products of three vectors summing to zero in a right triangle. This problem explores vector algebra, including the orthogonality of vectors and their dot product calculations. Suitable for Grades 10-12.

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