Problem 47 - A Graphical Approach to Precalculus with Limits

Question

Write each vector in the form ai \(+\) bj. Round a and b to the nearest hundredth, if necessary. $$\langle 2,0\rangle$$

Step-by-Step Solution

Answer

The vector is written as \( 2\mathbf{i} + 0\mathbf{j} \).

Step 1: Understanding Vectors

A vector has two components: an x-component and a y-component. In this exercise, the vector is \( \langle 2, 0 \rangle \), which means it has an x-component of 2 and a y-component of 0.

Step 2: Expressing in Component Form

Vectors in two dimensions can be expressed in terms of \( \mathbf{i} \) and \( \mathbf{j} \), where \( \mathbf{i} \) is the unit vector along the x-axis and \( \mathbf{j} \) is the unit vector along the y-axis. Given the vector \( \langle 2, 0 \rangle \), it can be expressed in this form.

Step 3: Writing in Terms of i and j

The vector \( \langle 2, 0 \rangle \) can be written as \( 2\mathbf{i} + 0\mathbf{j} \). This expresses the vector as a sum of its components along the x-axis and y-axis.

Step 4: Rounding Components

The problem asks for rounding to the nearest hundredth if necessary. Since both components (2 for \( \mathbf{i} \) and 0 for \( \mathbf{j} \)) are already integers, no rounding is needed.

Key Concepts

Components of Vectors

Vectors are not just abstract mathematical objects. They are essential in representing quantities that have both magnitude and direction. When tackling vectors, it's crucial to break them down into parts that are easy to handle, known as components. Let's look into these components to understand how they work.

• X-component: Represents the influence of a vector in the horizontal direction. For instance, in the vector \( \langle 2, 0 \rangle \), the x-component is 2.


• Y-component: Represents the influence of a vector in the vertical direction. For the same vector, the y-component is 0.

Breaking down the vector in this manner makes it much simpler to apply mathematical operations like addition or multiplication. Each component can be treated individually, simplifying calculations and visual representations.

Unit Vectors

Unit vectors are incredibly handy tools when working with vectors. You can think of unit vectors as the building blocks of vector components. They provide a standardized way to express vector components in a consistent manner.

• Definition: A unit vector has a magnitude of 1 and merely points in a specific direction.


• X-axis: For the horizontal component, the unit vector is \( \mathbf{i} \).


• Y-axis: For the vertical component, the unit vector is \( \mathbf{j} \).

To express any vector, like \( \langle 2, 0 \rangle \), you simply multiply the magnitude of each component by these unit vectors. Thus, \( \langle 2, 0 \rangle \) translates to \( 2\mathbf{i} + 0\mathbf{j} \). This notation makes it easier to visualize and manipulate vectors in mathematical equations.

Rounding Numbers

Rounding numbers is a fundamental mathematical practice that simplifies numbers to make them easier to work with. This process becomes particularly useful in calculations where precision is needed up to a certain number of decimal places. When dealing with vector components, it's often necessary to round numbers to ensure clarity and uniformity.

• Nearest Hundredth: When asked to round to the nearest hundredth, you look at the third decimal place to decide how to round the second one.


• Vector Application: In the exercise with vector \( \langle 2, 0 \rangle \), no rounding was required as both components were integers. However, in more complex cases, rounding might be essential.

By rounding, you create a cleaner presentation, making calculations more straightforward without significantly losing accuracy.