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


<IMAGE> ​​

Key Concepts

Vector Notation

Vector notation is a mathematical representation of vectors, typically expressed in the form of ordered pairs or triples, such as u = <x, y> in two dimensions or u = <x, y, z> in three dimensions. This notation allows for clear communication of a vector's direction and magnitude, which is essential for performing vector operations like addition, subtraction, and scalar multiplication.

Magnitude of a Vector

The magnitude of a vector is a measure of its length, calculated using the Pythagorean theorem. For a vector u = <x, y>, the magnitude is given by ||u|| = √(x² + y²). Understanding how to compute the magnitude is crucial for analyzing vectors, especially when determining their relative sizes or when normalizing them to unit vectors.

Vector Components

Vector components refer to the projections of a vector along the axes of a coordinate system. For example, in a two-dimensional space, a vector u can be broken down into its horizontal (x) and vertical (y) components. This decomposition is vital for solving problems involving vectors, as it simplifies calculations and helps in visualizing vector addition and other operations.