https://r2.erweima.ai/mathbot/image/17261971508076uwain3v.jpg

Let's break down and solve the problems in the image.

### Problem 1: Finding the Angle Between Vectors \( \mathbf{A} \) and \( \mathbf{B} \)
You are given two vectors:
- \( \mathbf{A} = \mathbf{D} + \mathbf{T} \)
- \( \mathbf{B} = 2i + 2j + k \)

With the magnitudes:
- \( D = 16 \) and \( T = 36 \).

We need to find the angle between vectors \( \mathbf{A} \) and \( \mathbf{B} \).

1. **Components of \( \mathbf{A} \):**
   \( \mathbf{A} = \mathbf{D} + \mathbf{T} \), where:
   - \( \mathbf{D} \) has a magnitude of 16 and direction given by \( 50^\circ \) relative to the x-axis.
   - \( \mathbf{T} \) has a magnitude of 36 and direction of \( 22^\circ \) relative to the y-axis.

   Using trigonometry to resolve \( \mathbf{D} \) and \( \mathbf{T} \) into components:

   - For \( \mathbf{D} \):
     \[
     D_x = 16 \cos(50^\circ), \quad D_y = 16 \sin(50^\circ)
     \]
   - For \( \mathbf{T} \):
     \[
     T_x = 36 \sin(22^\circ), \quad T_y = 36 \cos(22^\circ)
     \]

   Therefore, \( \mathbf{A} = (D_x + T_x)\hat{i} + (D_y + T_y)\hat{j} \).

2. **Angle Between \( \mathbf{A} \) and \( \mathbf{B} \):**
   The formula for the angle \( \theta \) between two vectors is given by:
   \[
   \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|}
   \]
   We first need to calculate \( \mathbf{A} \cdot \mathbf{B} \) and the magnitudes of \( \mathbf{A} \) and \( \mathbf{B} \).

### Problem 2: Circular Motion of a Car
A car with wheels of radius 33 cm starts from rest, accelerates for 4 seconds until its wheels reach an angular velocity of 31.5 rad/s, then decelerates over 6 seconds until stopping. You need to find the total distance traveled by the car and the total angle rotated by the wheels.

### Problem 3: Static Equilibrium of a System
Given two masses \( A \) (13 kg) and \( B \) (7 kg), with a friction coefficient of 0.2 between \( A \) and the table, find the minimum mass of \( C \) to prevent motion and calculate the acceleration if mass \( C \) is removed.

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I'll begin solving these problems step by step.

**Would you like me to solve one of them first or go step-by-step through all three?**

Encuentre el ángulo entre los vectores A y B, calcule la distancia total recorrida por un automóvil en movimiento circular, y determine la masa mínima de C en un sistema de equilibrio estático.

This page covers a series of physics problems including vector angle calculation using dot product, determining total distance traveled by a car in circular motion, and finding the minimum mass required for static equilibrium in a system with friction.

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