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 \( \mathbf{u} \) and \( \mathbf{v} \) explicitly. If the figure provides magnitudes and directions (angles), 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. \[ \mathbf{u} + \mathbf{v} = (u_x + v_x, \; u_y + v_y) \]
5. Express the resultant vector \( \mathbf{u} + \mathbf{v} \) in vector notation, typically as \( \langle u_x + v_x, \; u_y + v_y \rangle \).
6. If needed, find the magnitude and direction of the resultant vector using:
7. \[ |\mathbf{u} + \mathbf{v}| = \sqrt{(u_x + v_x)^2 + (u_y + v_y)^2} \]
8. and
9. \[ \theta = \tan^{-1} \left( \frac{u_y + v_y}{u_x + v_x} \right) \]

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 expresses vectors typically as ordered pairs or triplets (e.g., <x, y>) representing their components along coordinate axes. This notation simplifies calculations and clearly shows the direction and magnitude of vectors.

Graphical Representation of Vectors

Vectors can be represented graphically as arrows with direction and length proportional to magnitude. Understanding how to interpret and draw vectors on a coordinate plane is essential for visualizing vector addition and verifying results.