Use the figure to find each vector: u - v. Use vector notation as in Example 4.


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Verified step by step guidance

1. Step 1: Identify the components of vector \( \mathbf{u} \) and vector \( \mathbf{v} \) from the given figure.
2. Step 2: Write vector \( \mathbf{u} \) in component form as \( \mathbf{u} = \langle u_1, u_2 \rangle \).
3. Step 3: Write vector \( \mathbf{v} \) in component form as \( \mathbf{v} = \langle v_1, v_2 \rangle \).
4. Step 4: Subtract the components of vector \( \mathbf{v} \) from vector \( \mathbf{u} \) to find \( \mathbf{u} - \mathbf{v} = \langle u_1 - v_1, u_2 - v_2 \rangle \).
5. Step 5: Express the result in vector notation as \( \langle u_1 - v_1, u_2 - v_2 \rangle \).

Key Concepts

Vector Notation

Vector notation is a way to represent vectors in a mathematical format, typically using angle brackets. For example, a vector u can be expressed as u = <u1, u2>, where u1 and u2 are its components along the x and y axes. Understanding this notation is essential for performing vector operations, such as addition and subtraction.

Vector Subtraction

Vector subtraction involves finding the difference between two vectors, which can be visualized geometrically. If u = <u1, u2> and v = <v1, v2>, then the subtraction u - v is calculated as <u1 - v1, u2 - v2>. This operation results in a new vector that represents the direction and magnitude from the tip of vector v to the tip of vector u.

Geometric Interpretation of Vectors

Vectors can be represented graphically as arrows in a coordinate system, where the length indicates magnitude and the direction shows orientation. This geometric interpretation helps in visualizing operations like addition and subtraction, as it allows one to see how vectors interact in space. Understanding this concept is crucial for solving problems involving vector operations.