91Ó°ÊÓ

Vector addition is a fundamental operation that combines two or more vectors to produce a new vector. It involves adding the components of each vector separately. For instance, suppose we have two vectors in two-dimensional space, \( \mathbf{u} \) and \( \mathbf{v} \), with \( \mathbf{u} = (u_1, u_2) \) and \( \mathbf{v} = (v_1, v_2) \). Then their sum \( \mathbf{u} + \mathbf{v} \) is given by adding the corresponding components: \[ (u_1 + v_1, u_2 + v_2) \]. An example from our original exercise is adding \( \mathbf{u} = (1, 1) \) and \( \mathbf{v} = (1, -1) \). Their sum is \( (1 + 1, 1 - 1) = (2, 0) \).

• This sum represents a vector pointing in the positive x-direction, having zero y-component.
• Visualizing this on a graph shows that the vectors \( \mathbf{i} \) (x-axis) and \( \mathbf{j} \) (y-axis) help move the sum vector to its resulted destination in the plane.

Scalar multiplication is the process of multiplying a vector by a single number, known as a scalar. If you have a vector \( \mathbf{u} = (u_1, u_2) \) and a scalar \( c \), the resulting vector after scalar multiplication, \( c\mathbf{u} \), is calculated by multiplying each component of the vector by the scalar: \[ c\mathbf{u} = (c \times u_1, c \times u_2) \]. In our exercise, we see this concept in practice with calculations like \( 2\mathbf{u} \) and \( -3\mathbf{v} \):

• For \( 2\mathbf{u} \), multiply each component of \( \mathbf{u} = (1, 1) \) by 2, resulting in \( (2, 2) \).
• For \( -3\mathbf{v} \), multiply each component of \( \mathbf{v} = (1, -1) \) by -3, resulting in \( (-3, 3) \).

The resulting vectors scale in magnitude but maintain or reverse direction depending on whether the scalar is positive or negative.

Vector components are key elements that define a vector with respect to a coordinate system. They express how much influence a vector has in each dimension, for example along the x-axis and y-axis. Consider a vector \( \mathbf{u} = \mathbf{i} + \mathbf{j} \). Here, \( \mathbf{i} \) and \( \mathbf{j} \) are standard unit vectors pointing along the axes:

• \( \mathbf{i} = (1, 0) \) represents a step of one unit along the x-axis.
• \( \mathbf{j} = (0, 1) \) represents a step of one unit along the y-axis.

The notation \( (1, 1) \) for \( \mathbf{u} \) implies a combination of both these unit vectors. This means \( \mathbf{u} \) moves one unit right and one unit up in the 2D plane. Vector \( \mathbf{v} = \mathbf{i} - \mathbf{j} \) translates to components \( (1, -1) \), with one unit step right and one unit down, effectively showing the influence in each direction. In a broader sense, vector components allow us to analyze and manipulate vectors by understanding their behavior in each direction independently.