Given vectors u and v, find: 2u + 3v. 
u = 〈-1, 2〉, v = 〈3, 0〉

Verified step by step guidance

1. Identify the components of vector \( \mathbf{u} \) as \( \langle -1, 2 \rangle \) and vector \( \mathbf{v} \) as \( \langle 3, 0 \rangle \).
2. Multiply each component of vector \( \mathbf{u} \) by 2 to get \( 2\mathbf{u} = \langle 2(-1), 2(2) \rangle \).
3. Multiply each component of vector \( \mathbf{v} \) by 3 to get \( 3\mathbf{v} = \langle 3(3), 3(0) \rangle \).
4. Add the corresponding components of \( 2\mathbf{u} \) and \( 3\mathbf{v} \) to find \( 2\mathbf{u} + 3\mathbf{v} = \langle 2(-1) + 3(3), 2(2) + 3(0) \rangle \).
5. Simplify the expression to find the resulting vector.

Key Concepts

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 example, if u = 〈-1, 2〉 and v = 〈3, 0〉, the sum u + v is calculated by adding the first components (-1 + 3) and the second components (2 + 0), resulting in the vector 〈2, 2〉.

Scalar Multiplication

Scalar multiplication refers to the process of multiplying a vector by a scalar (a real number), which scales the vector's magnitude without changing its direction. For instance, multiplying vector u = 〈-1, 2〉 by the scalar 2 results in the vector 〈-2, 4〉, effectively doubling its length while maintaining its direction.

Linear Combinations of Vectors

A linear combination of vectors involves multiplying each vector by a scalar and then adding the results. In the given problem, the expression 2u + 3v represents a linear combination where vector u is scaled by 2 and vector v by 3. This results in a new vector that combines the effects of both original vectors, allowing for a wide range of applications in physics and engineering.