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 expressed in component form as \( \mathbf{u} = \langle u_x, u_y \rangle \) and \( \mathbf{v} = \langle v_x, v_y \rangle \), where \( u_x \) and \( u_y \) are the horizontal and vertical components of \( \mathbf{u} \), and similarly for \( \mathbf{v} \).
2. Write down the components of \( \mathbf{u} \) and \( \mathbf{v} \) explicitly based on the figure, noting their directions and magnitudes along the x- and y-axes.
3. Subtract the components of \( \mathbf{v} \) from the components of \( \mathbf{u} \) to find \( \mathbf{u} - \mathbf{v} \). This is done component-wise: \( \mathbf{u} - \mathbf{v} = \langle u_x - v_x, u_y - v_y \rangle \).
4. Express the resulting vector \( \mathbf{u} - \mathbf{v} \) in vector notation, using angle brackets and the components you calculated.
5. If needed, verify your result by sketching the vectors \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{u} - \mathbf{v} \) to ensure the subtraction aligns with the geometric interpretation of vector subtraction.

Key Concepts

Vector Representation and Notation

Vectors are quantities with both magnitude and direction, often represented as directed line segments or coordinate pairs. Vector notation typically uses angle brackets, e.g., u = <x, y>, to denote components along the x and y axes. Understanding this notation is essential for performing vector operations like addition and subtraction.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors by subtracting their corresponding components. If u = <u_x, u_y> and v = <v_x, v_y>, then u - v = <u_x - v_x, u_y - v_y>. This operation results in a new vector representing the displacement from v to u.

Using Geometric Interpretation of Vectors

Vectors can be visualized geometrically as arrows in the plane. Subtracting vectors corresponds to adding the negative of a vector, which can be interpreted as reversing the direction of v and then adding it to u. This geometric view helps in understanding vector operations beyond algebraic manipulation.