Problem 51 - Precalculus

Question

Perform the indicated vector operation. $$(5 i-2 j)+(-3 i+2 j)$$

Step-by-Step Solution

Answer

The result of the vector operation is \(2i\).

Step 1: Identify the vectors

In this expression, we have two vectors \(5i - 2j\) and \(-3i + 2j\). Each vector is composed of terms along the \(i\)-axis (horizontal) and \(j\)-axis (vertical).

Step 2: Combine the i components

Add the \(i\) components of both vectors. The \(i\) components are \(5i\) and \(-3i\). Their sum is: \(5i + (-3i) = 2i\).

Step 3: Combine the j components

Add the \(j\) components of both vectors. The \(j\) components are \(-2j\) and \(2j\). Their sum is: \(-2j + 2j = 0j\), which simplifies to zero.

Step 4: Write the final vector

Combine the results from the \(i\) and \(j\) components to form the final vector: \(2i + 0j = 2i\).

Key Concepts

i and j components

In the world of vectors, we often break down two-dimensional vectors into simpler parts called components. These components help us understand the direction and magnitude of the vector. The i and j components represent the horizontal and vertical directions, respectively.

An expression like \(5i - 2j\) is made up of two parts:

• \(5i\), which points in the horizontal direction.
• \(-2j\), which points in the vertical direction.

The term i indicates that we are moving along the x-axis (or horizontally), while the term j indicates movement along the y-axis (or vertically). This breakdown into i and j components makes it easier to add, subtract, or compare vectors.

component-wise addition

Adding vectors involves combining their respective components. This process is known as component-wise addition. It is a step-by-step process that ensures each corresponding part of the vectors is accounted for. Let’s see how it’s done using the example \((5i - 2j) + (-3i + 2j)\).

First, identify the i components from both vectors:

• From \(5i - 2j\), we take \(5i\).
• From \(-3i + 2j\), we take \(-3i\).

Add these i components together: \(5i + (-3i) = 2i\).

Next, do the same with the j components:

• From \(5i - 2j\), we take \(-2j\).
• From \(-3i + 2j\), we take \(2j\).

Add these j components: \(-2j + 2j = 0j\), which simplifies to zero.

The overall vector, formed by adding these components, becomes \(2i + 0j\) or simply \(2i\). Component-wise addition helps us add vectors in a systematic and accurate way.

vector operations

Vector operations include many mathematical processes like addition, subtraction, and scaling. Understanding these operations helps in solving various physics and engineering problems. Let's focus on vector addition, one of the fundamental vector operations.

In our example, \((5i - 2j) + (-3i + 2j)\), vector addition requires us to combine the corresponding i and j components of each vector. This is done by:

• Adding the values of the i components from both vectors.
• Adding the values of the j components in a similar manner.

Once you add these components, they form a new resultant vector based on their combined horizontal and vertical effects.

It is important to remember that these vector operations follow basic algebra rules and require careful attention to signs (positive or negative). Understanding these operations is crucial for accurately solving problems that involve vectors in real-world scenarios.