Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
- 2u + 4v

Verified step by step guidance

1. Identify the given vectors: \( \mathbf{u} = \langle -2, 5 \rangle \) and \( \mathbf{v} = \langle 4, 3 \rangle \).
2. Multiply the vector \( \mathbf{u} \) by 2: \( 2\mathbf{u} = 2 \times \langle -2, 5 \rangle = \langle 2 \times -2, 2 \times 5 \rangle \).
3. Multiply the vector \( \mathbf{v} \) by 4: \( 4\mathbf{v} = 4 \times \langle 4, 3 \rangle = \langle 4 \times 4, 4 \times 3 \rangle \).
4. Add the resulting vectors from the previous steps: \( 2\mathbf{u} + 4\mathbf{v} = \langle 2 \times -2, 2 \times 5 \rangle + \langle 4 \times 4, 4 \times 3 \rangle \).
5. Combine the corresponding components of the vectors: \( \langle (2 \times -2) + (4 \times 4), (2 \times 5) + (4 \times 3) \rangle \).

Key Concepts

Vector Operations

Vector operations involve mathematical procedures applied to vectors, such as addition, subtraction, and scalar multiplication. In this case, we are performing scalar multiplication on vectors u and v, which means multiplying each component of the vectors by a scalar value. Understanding how to manipulate vectors is essential for solving problems involving vector quantities.

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a scalar (a single number), which scales the vector's magnitude without changing its direction. For example, multiplying vector u = 〈-2, 5〉 by -2 results in a new vector that points in the opposite direction and has a length twice that of u. This concept is crucial for calculating expressions like -2u in the given problem.

Vector Addition

Vector addition involves combining two or more vectors to produce a resultant vector. This is done by adding the corresponding components of the vectors. For instance, if we have two vectors u and v, their sum is calculated by adding their x-components and y-components separately. This concept is necessary for finding the final result of the expression -2u + 4v.