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

Verified step by step guidance

1. Identify the vector \( \mathbf{u} \) from the figure, noting its direction and magnitude relative to the coordinate axes or reference points given.
2. Recall that the negative of a vector \( \mathbf{u} \), denoted \( -\mathbf{u} \), has the same magnitude as \( \mathbf{u} \) but points in the exact opposite direction.
3. Express the vector \( \mathbf{u} \) in component form, typically as \( \mathbf{u} = \langle u_x, u_y \rangle \), where \( u_x \) and \( u_y \) are the horizontal and vertical components respectively.
4. To find \( -\mathbf{u} \), multiply each component of \( \mathbf{u} \) by \( -1 \), resulting in \( -\mathbf{u} = \langle -u_x, -u_y \rangle \).
5. Write the final answer in vector notation, clearly indicating the components of \( -\mathbf{u} \) as derived from the previous step.

Key Concepts

Vector Notation

Vector notation represents vectors using components along coordinate axes, typically written as ⟨x, y⟩. This concise form shows the horizontal and vertical parts of a vector, making it easier to perform operations like addition or scalar multiplication.

Vector Components and Direction

A vector's components correspond to its projections on the coordinate axes, determined by its magnitude and direction. Understanding how to resolve a vector into horizontal and vertical parts using trigonometric functions is essential for accurate representation.

Using Reference Figures for Vectors

Interpreting vectors from a figure involves identifying their initial and terminal points, direction, and length. This visual information helps translate the vector into component form, which is necessary for expressing it in vector notation.