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. Typically, vectors are represented in the form \( \langle a, b \rangle \), where \( a \) is the horizontal component and \( b \) is the vertical component.
2. Write down the components of vector \( \mathbf{u} \) as \( \langle u_1, u_2 \rangle \) and vector \( \mathbf{v} \) as \( \langle v_1, v_2 \rangle \).
3. To find the resultant vector \( \mathbf{u} + \mathbf{v} \), add the corresponding components of the vectors: \( \langle u_1 + v_1, u_2 + v_2 \rangle \).
4. Substitute the values of \( u_1, u_2, v_1, \) and \( v_2 \) from the figure into the expression for the resultant vector.
5. Express the final vector \( \mathbf{u} + \mathbf{v} \) in vector notation as \( \langle \text{sum of horizontal components}, \text{sum of vertical components} \rangle \).

Key Concepts

Vector Addition

Vector addition involves combining two or more vectors to form a resultant vector. This is typically done by adding the corresponding components of the vectors. For example, if vector u has components (u1, u2) and vector v has components (v1, v2), then the sum u + v is given by (u1 + v1, u2 + v2). Understanding this concept is crucial for solving problems involving multiple vectors.

Vector Notation

Vector notation is a way to represent vectors in a mathematical format, often using angle brackets or boldface letters. For instance, a vector can be denoted as u = <u1, u2> or in bold as **u**. This notation helps in clearly distinguishing vectors from scalar quantities and is essential for performing operations like addition, subtraction, and scalar multiplication.

Graphical Representation of Vectors

Graphical representation of vectors involves illustrating vectors as arrows in a coordinate system, where the length represents the magnitude and the direction indicates the vector's direction. This visual approach aids in understanding vector operations, such as addition, by allowing one to see how vectors combine geometrically. It is particularly useful when working with problems that involve angles and directions.