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 \( \mathbf{u} \), respectively, and similarly for \( \mathbf{v} \).
2. Write down the subtraction operation for vectors: \( \mathbf{u} - \mathbf{v} = \langle u_x - v_x, u_y - v_y \rangle \). This means you subtract the corresponding components of \( \mathbf{v} \) from \( \mathbf{u} \).
3. Calculate the difference of the horizontal components: subtract \( v_x \) from \( u_x \) to find the \( x \)-component of \( \mathbf{u} - \mathbf{v} \).
4. Calculate the difference of the vertical components: subtract \( v_y \) from \( u_y \) to find the \( y \)-component of \( \mathbf{u} - \mathbf{v} \).
5. Express the resulting vector in vector notation as \( \mathbf{u} - \mathbf{v} = \langle u_x - v_x, u_y - v_y \rangle \). This is the vector difference you were asked to find.

Key Concepts

Vector Subtraction

Vector subtraction involves finding the difference between two vectors by reversing the direction of the vector to be subtracted and then adding it to the first vector. Algebraically, u - v is equivalent to u + (-v), where -v is the vector v with its direction reversed.

Vector Notation

Vector notation typically represents vectors as ordered pairs or components, such as u = <x, y>. This notation allows for straightforward arithmetic operations like addition and subtraction by working component-wise, which is essential for expressing the result of u - v clearly.

Graphical Representation of Vectors

Vectors can be represented graphically as directed line segments with magnitude and direction. Understanding how to visualize vector subtraction on a graph helps in interpreting the problem and verifying the algebraic result by drawing vectors u, v, and u - v accordingly.