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. Identify the components of vector \( \mathbf{u} \) and vector \( \mathbf{v} \) from the given figure.
2. Express vector \( \mathbf{u} \) in component form as \( \langle u_1, u_2 \rangle \).
3. Express vector \( \mathbf{v} \) in component form as \( \langle v_1, v_2 \rangle \).
4. Subtract the components of vector \( \mathbf{v} \) from vector \( \mathbf{u} \) to find \( \mathbf{u} - \mathbf{v} \), resulting in \( \langle u_1 - v_1, u_2 - v_2 \rangle \).
5. Write the resulting vector \( \mathbf{u} - \mathbf{v} \) 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 is done by subtracting their corresponding components. If u = <u1, u2> and v = <v1, v2>, then the vector u - v is calculated as <u1 - v1, u2 - v2>. This concept is crucial for solving problems that require the determination of relative positions or directions.

Geometric Interpretation of Vectors

Vectors can be visually represented as arrows in a coordinate system, where the direction indicates the vector's orientation and the length represents its magnitude. This geometric interpretation helps in understanding operations like subtraction, as it can be visualized as moving from the tip of one vector to the tip of another. Recognizing this relationship aids in comprehending the spatial aspects of vector operations.