Problem 47 - Precalculus

Question

Express the vector in terms of unit vectors i and \(j\) $$(5,-3)$$

Step-by-Step Solution

Answer

The vector is expressed as \(5\mathbf{i} - 3\mathbf{j}\).

Step 1: Understanding the Vector

The vector given is \((5, -3)\). This represents a two-dimensional vector, where \(5\) is the component along the x-axis and \(-3\) is the component along the y-axis.

Step 2: Expressing X Component

The x-component, \(5\), is represented in terms of the unit vector \(\mathbf{i}\), which is the unit vector along the x-axis. So, this becomes \(5\mathbf{i}\).

Step 3: Expressing Y Component

The y-component, \(-3\), is represented in terms of the unit vector \(\mathbf{j}\), which is the unit vector along the y-axis. So, this becomes \(-3\mathbf{j}\).

Step 4: Combining Both Components

Combine both components together to express the complete vector in terms of the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\). The vector is expressed as \(5\mathbf{i} - 3\mathbf{j}\).

Key Concepts

Unit Vector

A unit vector is a vector with a magnitude of 1. It is used to indicate direction, without considering the vector's length. In a two-dimensional coordinate system, we often use two specific unit vectors:

• The unit vector \( \mathbf{i} \), pointing along the x-axis.
• The unit vector \( \mathbf{j} \), pointing along the y-axis.

These unit vectors are foundational in expressing other vectors. By scaling the unit vectors, you can represent any vector in terms of its components. When we say a component is along \( \mathbf{i} \) or \( \mathbf{j} \), it simply means the direction is along x or y respectively. Vectors expressed in this manner make it easier to understand and manipulate them for operations like addition or subtraction.

X and Y Components

In a two-dimensional vector, each part of the vector is associated with one of the coordinate axes. This is what we refer to as the components of the vector:

• The x-component, which aligns with the x-axis.
• The y-component, which aligns with the y-axis.

Each of these components is a projection of the original vector along the respective axis. For example, in the vector \((5, -3)\), the x-component is 5, while the y-component is -3. When expressing the entirety of the vector, every component is multiplied by its respective unit vector and added together. This expresses the vector concisely and aligns it with the coordinate system, allowing the vector to be written in the form \(5\mathbf{i} - 3\mathbf{j}\). This formatting emphasizes that each component has both magnitude and direction within the plane.

Two-Dimensional Vector

A two-dimensional vector is a vector that exists on a flat plane, specified by two numerical values: its x and y components. These components define not just how far a point lies from the origin, but in which direction from the origin it extends.

Key Characteristics:

• It has both magnitude and direction. The magnitude can be calculated using the Pythagorean theorem: \( \sqrt{x^2 + y^2} \).
• The direction is often expressed in terms of unit vectors to show how much the vector points along the x and y axes.

Understanding two-dimensional vectors is crucial because they form the basis for more complex vector operations and applications in fields like engineering, physics, and computer graphics. A vector like \((5, -3)\) clearly demonstrates the balance and interaction between its components, laying the groundwork for many practical applications.