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The problem appears to involve three vectors in a right triangle, with the condition that their sum equals zero. We are asked to calculate the dot products of vector pairs \(\vec{A} \cdot \vec{B}\), \(\vec{A} \cdot \vec{C}\), and \(\vec{B} \cdot \vec{C}\).

From the image, the triangle is a right triangle with sides labeled 3, 4, and 5, which suggests it is a Pythagorean triple. This means that we can assume the magnitudes of the vectors correspond to these lengths, and the vectors sum to zero. Let's outline the key steps to solve this problem:

### Step 1: Identify vectors
- The vectors correspond to the sides of the triangle:
  - \(\vec{A}\) can be aligned with the side of length 4 (adjacent to the angle).
  - \(\vec{B}\) can be aligned with the side of length 3 (opposite to the angle).
  - \(\vec{C}\) is the hypotenuse, with length 5.

### Step 2: Assign coordinates
Since this is a right triangle:
- \(\vec{A}\) could have coordinates (4, 0) since it lies along the horizontal axis.
- \(\vec{B}\) could have coordinates (0, 3) since it lies along the vertical axis.
- \(\vec{C}\) could have coordinates (-4, -3) because \(\vec{A} + \vec{B} + \vec{C} = 0\).

### Step 3: Calculate dot products
We calculate each dot product using the formula \(\vec{A} \cdot \vec{B} = A_xB_x + A_yB_y\).

1. \(\vec{A} \cdot \vec{B}\):
   \[
   \vec{A} = (4, 0), \quad \vec{B} = (0, 3)
   \]
   \[
   \vec{A} \cdot \vec{B} = 4 \cdot 0 + 0 \cdot 3 = 0
   \]

2. \(\vec{A} \cdot \vec{C}\):
   \[
   \vec{A} = (4, 0), \quad \vec{C} = (-4, -3)
   \]
   \[
   \vec{A} \cdot \vec{C

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

Solve for the dot products of three vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) in a right triangle using vector algebra. Learn the step-by-step process, including applying the dot product formula and the Pythagorean theorem. Suitable for students in Grades 9-12.

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