Problem 46 - Precalculus

Question

Express the vector in terms of unit vectors i and \(j\) $$(-2,4)$$

Step-by-Step Solution

Answer

The vector is \(-2\mathbf{i} + 4\mathbf{j}\).

Step 1: Identify Components

The vector given is \((-2, 4)\), where the first number represents the x-component and the second number represents the y-component.

Step 2: Write Components in Unit Vector Form

The unit vector \(\mathbf{i}\) corresponds to the x-component, and \(\mathbf{j}\) corresponds to the y-component. Therefore, the x-component \(-2\) can be written as \(-2\mathbf{i}\), and the y-component \(4\) can be written as \(4\mathbf{j}\).

Step 3: Combine Unit Vector Components

Combine the expressions from the previous step to express the vector in terms of unit vectors: \((-2\mathbf{i} + 4\mathbf{j})\).

Key Concepts

Understanding Unit Vectors

Imagine a unit vector as a direction pointer, showing the way. In terms of mathematics, a unit vector has a magnitude of 1. This means that the length of the unit vector is always 1, which helps in maintaining direction without altering size. Unit vectors are essential because they allow you to express other vectors in terms of direction. Think of it as using a compass needle to point north without needing to worry about how far north you're going. To get a unit vector, you take any vector and divide each of its components by the vector's magnitude. The unit vectors for the Cartesian coordinate system are particularly important:

• The unit vector in the x-direction is usually represented by \(\mathbf{i}\).
• The unit vector in the y-direction is represented by \(\mathbf{j}\).

These vectors are used for defining any vector in the 2-dimensional plane. Keep in mind that unit vectors help you break down and understand the fundamental components of more complex vectors. It's like translating a language into its basic elements so you can understand it piece by piece.

Exploring Vector Components

Every vector can be broken down into its components, which are parts that help represent the vector along each axis. In a 2-D space, any vector can be described using its x-component and y-component.For example, the vector \((-2,4)\) consists of:

• -2 as the x-component, moving in the negative x-direction.
• 4 as the y-component, moving positively along the y-axis.

When we look at vector components, we're essentially looking at how much a vector moves in each direction. This process is similar to how you might describe a trip by saying you walked five blocks north and then two blocks west.By representing vectors as a sum of their components, you can easily visualize these moves as separate actions along the coordinate axes. This separation is crucial for calculations, as it allows you to add, subtract, and even compare vectors by handling simpler one-dimensional tasks.

The Role of i and j Notation

In physics and math, \(\mathbf{i}\) and \(\mathbf{j}\) notation offers a compact and efficient way to express 2-D vectors. Instead of writing out each vector component separately, \(\mathbf{i}\) and \(\mathbf{j}\) allow you to easily illustrate vectors by combining the components.When dealing with the vector \((-2, 4)\), using \(\mathbf{i}\) and \(\mathbf{j}\) simplifies it to \(-2\mathbf{i} + 4\mathbf{j}\). Each part of this expression tells you:

• The \(-2\mathbf{i}\) component refers to a movement of -2 units in the x-direction.
• The \(4\mathbf{j}\) component refers to a movement of 4 units in the y-direction.

By employing \(\mathbf{i}\) and \(\mathbf{j}\), you transform the components into a single, beautiful expression. It provides a clear, structured representation that highlights the vector's direction and magnitude components separately. This approach not only aids in visualizing vectors efficiently but also streamlines arithmetic operations on vectors such as addition and subtraction.