To prove the magnitude of the sum of two vectors, let's denote the vectors as **A** and **B**, with magnitudes **a** and **b** respectively, and they form an angle **θ** with each other when placed tail to tail. We will analyze the situation using components along two perpendicular axes, typically the x-axis and y-axis.

Breaking Down the Vectors into Components

First, we need to express the vectors **A** and **B** in terms of their components. For this, we can use trigonometric functions to resolve each vector into its x and y components.

Components of Vector A

Vector **A**, with magnitude **a**, can be resolved as follows:

• **A_x = a * cos(θ)** (the x-component)
• **A_y = a * sin(θ)** (the y-component)

Components of Vector B

Vector **B**, with magnitude **b**, will have its components resolved relative to the same axes. Since it makes an angle **θ** with **A**, the angle it makes with the x-axis will be **(180° - θ)**, assuming we are measuring angles in a counterclockwise direction:

• **B_x = b * cos(180° - θ) = -b * cos(θ)**
• **B_y = b * sin(180° - θ) = b * sin(θ)**

Summing the Components

Now, we can find the total components of the resultant vector **R** by adding the respective components of **A** and **B**:

• **R_x = A_x + B_x = a * cos(θ) - b * cos(θ) = (a - b) * cos(θ)**
• **R_y = A_y + B_y = a * sin(θ) + b * sin(θ) = (a + b) * sin(θ)**

Calculating the Magnitude of the Resultant Vector

To find the magnitude of the resultant vector **R**, we can use the Pythagorean theorem:

**|R| = √(R_x² + R_y²)**

Substituting our expressions for **R_x** and **R_y**:

**|R| = √(((a - b) * cos(θ))² + ((a + b) * sin(θ))²)**

Expanding and Simplifying

Now let's expand and simplify this expression:

**|R| = √((a - b)² * cos²(θ) + (a + b)² * sin²(θ))**

This can be further simplified to:

**|R| = √((a² - 2ab + b²) * cos²(θ) + (a² + 2ab + b²) * sin²(θ))**

Now, factoring out common terms:

**|R| = √(a²(cos²(θ) + sin²(θ)) + b²(cos²(θ) + sin²(θ)) + 2ab(sin²(θ) - cos²(θ)))**

Using the Pythagorean Identity

Since we know that **cos²(θ) + sin²(θ) = 1**, we can simplify further:

**|R| = √(a² + b² + 2ab * sin²(θ) - 2ab * cos²(θ))**

Final Expression for Magnitude

Finally, the magnitude of the resultant vector can be expressed as:

**|R| = √(a² + b² + 2ab * cos(θ))**

This result is a powerful expression in vector addition, as it succinctly captures how two vectors combine based on their magnitudes and the angle between them. This concept is foundational in physics and engineering, particularly in understanding forces, velocities, and other vector quantities.