Problem 54 - Precalculus

Question

Perform the indicated vector operation. $$(i-3 j)-(-2 i+j)$$

Step-by-Step Solution

Answer

The resultant vector is \(3i - 4j\).

Step 1: Identify the Components

The given vector expression is \((i - 3j) - (-2i + j)\). First, identify the components of each vector. Here, \(i - 3j\) is the first vector and \(-2i + j\) is the second vector.

Step 2: Distribute the Negative Sign

Apply the negative sign to each component of the second vector. The expression becomes \((i - 3j) + (2i - j)\).

Step 3: Combine Like Terms

Add the corresponding components. For the \(i\) components: \(i + 2i = 3i\). For the \(j\) components: \(-3j - j = -4j\).

Step 4: Write the Resultant Vector

Combine the results to write the vector in the standard form. Thus, the resultant vector is \(3i - 4j\).

Key Concepts

Components of Vectors

Understanding the components of vectors is crucial when performing vector operations. Vectors can be broken down into their horizontal and vertical components, typically denoted using unit vectors like \(i\) and \(j\) in two-dimensional space. In the original exercise, the vector expression given is \((i - 3j) - (-2i + j)\). Here, \(i - 3j\) represents the first vector, with \(i\) as the horizontal component and \(-3j\) as the vertical component. Similarly, the vector \(-2i + j\) denotes another vector with \(-2i\) as its horizontal component and \(j\) as its vertical component. Recognizing these components is the first step in effectively handling any vector expression. This understanding helps you to properly prepare for subsequent operations, such as distributing signs or combining terms.

Distributing Negative Sign

After identifying vector components, the next key step is distributing any negative signs present in the expression. This step ensures that all vector components are correctly aligned for the operations that follow. In the problem \((i - 3j) - (-2i + j)\), a negative sign impacts the second vector \(-2i + j\). We must distribute this negative sign, which changes \(-2i + j\) into \((+2i - j)\).

• Convert \(-2i\) into \(2i\).
• Convert \(+j\) to \(-j\).

Now, the expression becomes \((i - 3j) + (2i - j)\). This clear visualization helps simplify the process for further steps like combining terms.

Combining Like Terms

Once vectors have been properly adjusted and prepared by distributing any negative signs, combining like terms comes next. This means adding up all the horizontal components together and then adding the vertical components separately. For the expression \((i - 3j) + (2i - j)\), we can:

• Add together the \(i\) components: \(i + 2i = 3i\).
• Add together the \(j\) components: \(-3j - j = -4j\).

Combining these like terms correctly will form a new vector consisting of these summed components. This is an essential part of vector operations as it consolidates the information into a simplified, more manageable format.

Resultant Vector

Finally, after combining like terms, we arrive at what is known as the resultant vector. This vector represents the culmination of all operations performed on the initial vectors. In our example, the resultant vector is \(3i - 4j\). This vector is a combination of the aggregated \(i\) and \(j\) components.

• The \(i\) component is the sum of all horizontal vector components, resulting in \(3i\).
• The \(j\) component is the sum of all vertical vector components, resulting in \(-4j\).

The resultant vector provides a final, simplified expression that describes the total influence of both vectors involved. It can be useful in further calculations or in graphical representations where vector diagrams are involved.