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 or its components if given.
2. Recall that the vector \( -\mathbf{u} \) is the vector \( \mathbf{u} \) reversed in direction but with the same magnitude.
3. If \( \mathbf{u} \) is given in component form as \( \mathbf{u} = \langle x, y \rangle \), then \( -\mathbf{u} = \langle -x, -y \rangle \).
4. If the vector \( \mathbf{u} \) is given graphically, determine its components by measuring or using trigonometric relationships based on the angle and length.
5. Write the vector \( -\mathbf{u} \) explicitly in vector notation, ensuring the direction is opposite to \( \mathbf{u} \) and the magnitude remains the same.

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 Direction and Magnitude

A vector has both magnitude (length) and direction. Understanding how to determine these from a figure is essential, as the vector's components correspond to its horizontal and vertical displacements, which define its direction and size.

Vector Operations (Negation)

Negating a vector reverses its direction while keeping its magnitude the same. If vector u = ⟨x, y⟩, then -u = ⟨-x, -y⟩. This concept is crucial when the question asks for -u, indicating the vector pointing opposite to u.