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. For example, \( \mathbf{u} = (u_x, u_y) \) and \( \mathbf{v} = (v_x, v_y) \).
2. Write down the components of each vector explicitly. If the figure provides magnitudes and directions, use trigonometric functions to find components: \( u_x = |\mathbf{u}| \cos \theta_u \), \( u_y = |\mathbf{u}| \sin \theta_u \), and similarly for \( \mathbf{v} \).
3. Add the corresponding components of the vectors to find the resultant vector \( \mathbf{u} + \mathbf{v} \):
4. \\
5. \[ \mathbf{u} + \mathbf{v} = (u_x + v_x, u_y + v_y) \]
6. Express the sum vector in vector notation, for example, \( \mathbf{u} + \mathbf{v} = \langle u_x + v_x, u_y + v_y \rangle \).
7. If needed, interpret the resulting vector by calculating its magnitude and direction using:
8. \[ |\mathbf{u} + \mathbf{v}| = \sqrt{(u_x + v_x)^2 + (u_y + v_y)^2} \]
9. and
10. \[ \theta = \tan^{-1} \left( \frac{u_y + v_y}{u_x + v_x} \right) \]
11. This completes the process of finding \( \mathbf{u} + \mathbf{v} \) using vector notation.

Key Concepts

Vector Addition

Vector addition involves combining two vectors to produce a resultant vector. This is done by adding their corresponding components or by placing the tail of the second vector at the head of the first and drawing the resultant from the tail of the first to the head of the second.

Vector Notation

Vector notation typically represents vectors as ordered pairs or triplets, such as (x, y) in two dimensions. This notation clearly shows the components along each axis, facilitating operations like addition and subtraction.

Graphical Representation of Vectors

Vectors can be represented graphically as arrows in a coordinate plane, where the length indicates magnitude and the direction shows orientation. Understanding how to interpret and draw vectors graphically helps visualize operations like addition.