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, 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 two vectors to find the components of \( \mathbf{u} + \mathbf{v} \): \( (u_x + v_x, u_y + v_y) \). This is the vector sum in component form.
4. Express the resulting vector \( \mathbf{u} + \mathbf{v} \) in vector notation as \( \langle u_x + v_x, u_y + v_y \rangle \). This notation clearly shows the horizontal and vertical components of the sum.
5. Optionally, if needed, find the magnitude and direction of \( \mathbf{u} + \mathbf{v} \) using the formulas: magnitude \( = \sqrt{(u_x + v_x)^2 + (u_y + v_y)^2} \) and direction \( = \tan^{-1} \left( \frac{u_y + v_y}{u_x + v_x} \right) \).

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 triples, such as u = <x, y>, indicating their components along coordinate axes. This notation simplifies calculations and clearly shows the direction and magnitude of vectors.

Component-wise Addition of Vectors

When adding vectors in component form, add their corresponding x-components and y-components separately. For example, if u = <u_x, u_y> and v = <v_x, v_y>, then u + v = <u_x + v_x, u_y + v_y>, which yields the resultant vector.