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

Verified step by step guidance

1. Identify the vector \( \mathbf{v} \) from the figure, noting its direction and magnitude as given or implied.
2. Recall that the negative of a vector \( \mathbf{v} \), denoted \( -\mathbf{v} \), has the same magnitude as \( \mathbf{v} \) but points in the exact opposite direction.
3. Express the vector \( \mathbf{v} \) in component form if possible, for example, \( \mathbf{v} = \langle v_x, v_y \rangle \), where \( v_x \) and \( v_y \) are the horizontal and vertical components respectively.
4. To find \( -\mathbf{v} \), multiply each component of \( \mathbf{v} \) by \( -1 \), resulting in \( -\mathbf{v} = \langle -v_x, -v_y \rangle \).
5. Write the final answer in vector notation, clearly indicating the components of \( -\mathbf{v} \) as derived from the previous step.

Key Concepts

Vector Notation

Vector notation represents vectors using components along coordinate axes, typically written as ⟨x, y⟩ in two dimensions. This notation simplifies vector operations like addition, subtraction, and scalar multiplication by expressing vectors as ordered pairs or triples.

Vector Components and Direction

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

Using Trigonometry to Find Vector Components

Trigonometric functions such as sine and cosine relate a vector's angle to its components. By applying these functions to the vector's magnitude and direction angle, one can calculate the precise x and y components needed for vector notation.