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

Verified step by step guidance

1. Identify the given vectors: \( \mathbf{u} = 2\mathbf{i} \) and \( \mathbf{v} = \mathbf{i} + \mathbf{j} \).
2. Multiply vector \( \mathbf{u} \) by 2: calculate \( 2\mathbf{u} = 2 \times 2\mathbf{i} \).
3. Multiply vector \( \mathbf{v} \) by 3: calculate \( 3\mathbf{v} = 3 \times (\mathbf{i} + \mathbf{j}) \).
4. Add the resulting vectors from the previous two steps: \( 2\mathbf{u} + 3\mathbf{v} \).
5. Combine like terms (i.e., the \( \mathbf{i} \) components together and the \( \mathbf{j} \) components together) to express the final vector in terms of \( \mathbf{i} \) and \( \mathbf{j} \).

Key Concepts

Vector Representation in Component Form

Vectors can be expressed as sums of their components along standard unit vectors, typically i and j in two dimensions. For example, u = 2i means the vector has a component 2 along the x-axis and 0 along the y-axis. Understanding this form allows for straightforward vector addition and scalar multiplication.

Scalar Multiplication of Vectors

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). For instance, multiplying vector u by 2 scales its magnitude by 2 without changing its direction. This operation is essential for combining vectors with different weights, as in 2u + 3v.

Vector Addition

Vector addition is performed by adding corresponding components of the vectors. For example, adding vectors a = a₁i + a₂j and b = b₁i + b₂j results in (a₁ + b₁)i + (a₂ + b₂)j. This principle is used to find the resultant vector 2u + 3v by first scaling and then adding the vectors.