Use the figure to find each vector: u + v. Use vector notation as in Example 4.

Verified step by step guidance

1. Identify the components of vectors \( \mathbf{u} \) and \( \mathbf{v} \) from the figure. Typically, each vector can be broken down into its horizontal (x) and vertical (y) components, such as \( \mathbf{u} = (u_x, u_y) \) and \( \mathbf{v} = (v_x, v_y) \).
2. Write down the components of \( \mathbf{u} \) and \( \mathbf{v} \) explicitly. For example, if \( \mathbf{u} \) points 3 units right and 4 units up, then \( \mathbf{u} = (3, 4) \). Do the same for \( \mathbf{v} \).
3. Add the corresponding components of the two vectors to find \( \mathbf{u} + \mathbf{v} \). This means calculating \( (u_x + v_x, u_y + v_y) \).
4. Express the resulting vector \( \mathbf{u} + \mathbf{v} \) in vector notation, for example, \( \mathbf{u} + \mathbf{v} = (u_x + v_x, u_y + v_y) \).
5. If needed, you can also represent the vector \( \mathbf{u} + \mathbf{v} \) graphically by drawing it from the origin or the tail of \( \mathbf{u} \) to the head of \( \mathbf{v} \), confirming the addition visually.

Key Concepts

Vector Addition

Vector addition involves combining two vectors to form a resultant vector by adding their corresponding components or by placing them head-to-tail graphically. The sum vector u + v represents the combined effect of vectors u and v.

Vector Notation

Vector notation typically expresses vectors in component form, such as ⟨x, y⟩, where x and y are the horizontal and vertical components. This notation simplifies calculations and clearly represents the vector's direction and magnitude.

Resolving Vectors into Components

To add vectors accurately, each vector is broken down into horizontal and vertical components using trigonometric functions if angles are given. This allows for straightforward addition of components to find the resultant vector.