Given vectors u and v, find: 2u + 3v.
u = 2i, v = i + j

Verified step by step guidance

1. Identify the components of vector \( \mathbf{u} \) and vector \( \mathbf{v} \). Here, \( \mathbf{u} = 2\mathbf{i} \) and \( \mathbf{v} = \mathbf{i} + \mathbf{j} \).
2. Multiply vector \( \mathbf{u} \) by 2: \( 2\mathbf{u} = 2(2\mathbf{i}) = 4\mathbf{i} \).
3. Multiply vector \( \mathbf{v} \) by 3: \( 3\mathbf{v} = 3(\mathbf{i} + \mathbf{j}) = 3\mathbf{i} + 3\mathbf{j} \).
4. Add the results from the previous two steps: \( 2\mathbf{u} + 3\mathbf{v} = 4\mathbf{i} + (3\mathbf{i} + 3\mathbf{j}) \).
5. Combine like terms to simplify: \( 4\mathbf{i} + 3\mathbf{i} + 3\mathbf{j} = (4 + 3)\mathbf{i} + 3\mathbf{j} \).

Key Concepts

Vector Addition

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

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, if vector u = (u1, u2) is multiplied by a scalar k, the resulting vector is ku = (ku1, ku2). This concept is essential for manipulating vectors in expressions like 2u and 3v in the given problem.

Unit Vectors

Unit vectors are vectors with a magnitude of one, often used to indicate direction. In the context of the problem, the vectors u and v are expressed in terms of the standard unit vectors i and j, which represent the x and y directions, respectively. Understanding unit vectors helps in visualizing and performing operations on vectors in a coordinate system.