Problem 29 - Precalculus Mathematics for Calculus

Question

Write the given vector in terms of \(\mathbf{i}\) and \(\mathbf{j}\). $$\mathbf{u}=\langle 3,0\rangle$$

Step-by-Step Solution

Answer

The vector \( \mathbf{u} \) is \( 3\mathbf{i} + 0\mathbf{j} \).

Step 1: Identify Vector Components

The given vector \( \mathbf{u} = \langle 3, 0 \rangle \) consists of two components. The first component, 3, represents the horizontal direction along the \( x \)-axis and the second component, 0, represents the vertical direction along the \( y \)-axis.

Step 2: Express the Horizontal Component

The horizontal component of the vector, 3, is associated with the unit vector \( \mathbf{i} \). Therefore, in terms of \( \mathbf{i} \), this component can be written as \( 3\mathbf{i} \).

Step 3: Express the Vertical Component

The vertical component of the vector is 0, meaning it has no contribution along the \( y \)-axis. Therefore, it can be expressed as \( 0\mathbf{j} \) in terms of the unit vector \( \mathbf{j} \).

Step 4: Combine the Components

Combine the expressions for both components to write the vector \( \mathbf{u} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \). The expression becomes \( \mathbf{u} = 3\mathbf{i} + 0\mathbf{j} \).

Key Concepts

Unit Vectors

Unit vectors play an essential role in vector notation, serving as the building blocks for defining the direction of a vector. These vectors have a magnitude of one and are used to indicate direction in both two-dimensional and three-dimensional spaces. In the context of a 2D plane, unit vectors are typically denoted as \( \mathbf{i} \) and \( \mathbf{j} \).

• \( \mathbf{i} \): Represents the unit vector along the x-axis. Its coordinates are \( \langle 1, 0 \rangle \).
• \( \mathbf{j} \): Represents the unit vector along the y-axis with coordinates \( \langle 0, 1 \rangle \).

Although unit vectors have a magnitude of one, they are crucial in expressing any vector direction. Understanding and visualizing these unit vectors helps when breaking down any given vector into its horizontal and vertical components. This breakdown simplifies calculations and provides a clear geometric interpretation of the vector's direction.

Vector Components

Vector components are essential for understanding how vectors function and interact in space. A vector typically consists of two or more components representing different directions. For example, in two-dimensional space:

• The horizontal component measures the vector's extension along the x-axis.
• The vertical component measures the vector's elongation along the y-axis.

When describing a vector, like \( \mathbf{u} = \langle 3, 0 \rangle \), the first number (3) is the horizontal component and the second number (0) is the vertical component. The components can then be combined using unit vectors to express the vector more understandably, especially for calculations and graphical representation. By breaking down a vector into its components, we can easily manipulate and visualize these aspects individually, enabling precise analysis and operations.

i and j Notation

The \( \mathbf{i} \) and \( \mathbf{j} \) notation is a fundamental way to represent vectors, especially in two-dimensional spaces. This notation links each vector component with a correspondingly oriented unit vector, making it intuitive to express and compute vectors.
The notation works by associating the horizontal component with \( \mathbf{i} \) and the vertical component with \( \mathbf{j} \).

• For a vector like \( \mathbf{u} = \langle 3, 0 \rangle \), \( 3\mathbf{i} \) represents the horizontal movement.
• The \( 0\mathbf{j} \) indicates no vertical movement.

When we express a vector as \( \mathbf{u} = 3\mathbf{i} + 0\mathbf{j} \), it becomes clear at a glance which components affect which axes. This clarity makes \( \mathbf{i} \) and \( \mathbf{j} \) notation particularly useful in fields like physics and engineering where understanding directional forces and movements is critical. The simplicity of this system streamlines computations and helps to avoid errors, ensuring that each component is accurately factored into equations and graphical representations.