Problem 46 - 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. $$\langle 6,-3\rangle$$

Step-by-Step Solution

Answer

The vector is \(6i - 3j\).

Step 1: Understand the Vector Notation

The notation \(\langle 6, -3 \rangle\) represents a vector with components 6 in the horizontal (x) direction and -3 in the vertical (y) direction. To write it in the form \( ai + bj \), identify these components as \(a\) and \(b\).

Step 2: Assign Components to a and b

Assign \(a = 6\) and \(b = -3\). This directly corresponds to the horizontal and vertical influences in the vector form \(ai + bj\).

Step 3: Write in Vector Form \( ai + bj \)

Substitute \(a\) and \(b\) into the form \( ai + bj \), which gives us \( 6i - 3j \).

Step 4: Round Components (if necessary)

Check if rounding is necessary. In this case, since \(a\) and \(b\) are whole numbers, no rounding is needed.

Key Concepts

Vector Components

The concept of vector components is crucial for understanding vectors in mathematics. Vectors are not just simple numbers; they are quantities that have both magnitude and direction. When dealing with vectors, we often need to break them down into their horizontal and vertical parts, known as components.
In the example \( \langle 6, -3 \rangle \), the number 6 is the component along the x-axis (horizontal direction), while -3 is the component along the y-axis (vertical direction).

• Components tell us how far and in what direction a vector moves along each axis.
• The x-component moves left/right, while the y-component moves up/down.

Understanding these components helps us translate the physical world into mathematical expressions that are easier to work with.

Vector Form

Writing vectors in their standard mathematical form is an important skill. The standard form is often expressed as \( ai + bj \). Here, \( a \) represents the x-component and \( b \) the y-component of the vector. The letters \( i \) and \( j \) are unit vectors.

• \( i \) indicates a unit vector in the horizontal direction (x-axis).
• \( j \) signifies a unit vector in the vertical direction (y-axis).

For the vector \( \langle 6, -3 \rangle \), when written as \( 6i - 3j \), \( 6i \) shows movement along the x-axis, while \( -3j \) indicates movement along the y-axis.
This allows for a clear visual representation of vector components, making it easy to understand how a vector behaves.

Rounding Numbers

Rounding numbers is often needed when dealing with calculations in mathematics. It helps to keep numbers concise and easier to manage. For vectors, especially in form \( ai + bj \), rounding may come into play if the components are not whole numbers.

• Rounding to the nearest hundredth means looking at two decimal places.
• If the number in the third decimal place is 5 or more, round the second decimal place up by one.

In our example, the components were whole numbers, so no rounding was needed. However, knowing how to round helps when working with results that have decimals, ensuring precise and readable answers.

Mathematical Notation

Mathematical notation is a universal language for mathematicians. It provides a consistent way to write expressions and equations, ensuring communication is clear and universally understood. When working with vectors, notation like \( ai + bj \) becomes extremely useful.
Mathematical symbols and expressions, like \( i \) and \( j \), serve as shortcuts that hold a lot of meaning.

• Using notation helps avoid misunderstanding, ensuring everyone reads mathematical expressions the same way.
• This standardization is important in education, helping students understand and apply concepts uniformly.

For students learning vector notation, becoming familiar with these symbols is a stepping stone to more advanced topics in math and physics.