Problem 8 - Precalculus : Building Concepts and Connections

Question

Write each of the given vectors in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). $$\mathbf{v}=\langle 5,-3\rangle$$

Step-by-Step Solution

Answer

The vector \(\mathbf{v}=\langle 5,-3\rangle\) in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) is 5\(\mathbf{i}\) - 3\(\mathbf{j}\).

Step 1: Identifying the components of the vector

The vector \(\mathbf{v}\) can be written as \(\mathbf{v}=\langle 5,-3\rangle\). This means the vector has two components, the x-component is +5 and the y-component is -3.

Step 2: Writing the components in terms of unit vectors

Unit vectors in the x and y directions are represented by \(\mathbf{i}\) and \(\mathbf{j}\) respectively. So, the x-component can be written as 5\(\mathbf{i}\) and the y-component can be written as -3\(\mathbf{j}\). Thus, the vector is the sum of these two parts.

Step 3: Forming the vector using the unit direction vectors

Now we combine the two components to form our vector in terms of unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). So, vector \(\mathbf{v}\) is written as 5\(\mathbf{i}\) - 3\(\mathbf{j}\).

Key Concepts

Vector Components

Understanding vector components is key in breaking down vectors into their simplest form. Consider a vector as a way to describe movement or force in a particular direction. The components of a vector represent this movement along the fundamental axes in a coordinate system — usually the x-axis and y-axis.

For instance, the vector \( \mathbf{v}=\langle 5,-3\rangle \) from our exercise indicates two directional movements: 5 units in the x direction (horizontal) and -3 units in the y direction (vertical). Positive values indicate movement in the positive direction of the axis, while negative values indicate movement in the opposite direction. It's crucial to grasp that together, these values define the overall direction and magnitude of the vector.

Vector Notation

Vector notation is the language we use to describe vectors succinctly and precisely. In mathematics, we frequently notate a vector in component form, which shows the magnitude of movement along each axis separately, usually enclosed within angle brackets like so: \( \langle x, y \rangle \).

This not only simplifies the expression of the vector but also streamlines the process of performing vector operations, such as addition, subtraction, or even finding the dot product. For example, the vector \( \mathbf{v} \) in the exercise is represented by \( \mathbf{v}=\langle 5,-3\rangle \) indicating its precise direction through its components.

I and J Unit Vectors

The concept of \( \mathbf{i} \) and \( \mathbf{j} \) unit vectors simplifies the representation of vectors. \( \mathbf{i} \) is a unit vector pointing in the positive direction of the x-axis, while \( \mathbf{j} \) points in the positive direction of the y-axis. As the name 'unit' suggests, each of these vectors has a magnitude of 1.

Using these unit vectors, we can express any 2-dimensional vector as a combination of movements along the x and y directions. Converting \( \mathbf{v}=\langle 5,-3\rangle \) into unit vector form, we write 5 units along the x-axis as \( 5\mathbf{i} \) and -3 units along the y-axis as \( -3\mathbf{j} \). Combined, \( \mathbf{v} \) now becomes \( \mathbf{v} = 5\mathbf{i} - 3\mathbf{j} \), seamlessly integrating both the magnitude and direction of the vector into a practical and operational form.