Problem 56 - Precalculus

Question

Perform the indicated vector operation. $$(-2 i+j)+(2 i-4 j)$$

Step-by-Step Solution

Answer

The resulting vector is \(0i - 3j\).

Step 1: Identify Components of Each Vector

The first vector is \(-2i + j\). It has a horizontal component of \(-2\) and a vertical component of \(1\). The second vector is \(2i - 4j\). It has a horizontal component of \(2\) and a vertical component of \(-4\).

Step 2: Add Horizontal Components

Add the horizontal components of the vectors: \(-2 + 2 = 0\). This is the horizontal component of the resulting vector.

Step 3: Add Vertical Components

Add the vertical components of the vectors: \(1 - 4 = -3\). This is the vertical component of the resulting vector.

Step 4: Write the Resulting Vector

Combine the results from Steps 2 and 3 to form the resulting vector: \(0i - 3j\).

Key Concepts

Vector Components

Vectors can be broken down into different parts known as components. The main components are usually along the horizontal and vertical axes. When dissecting a vector, you're looking at how much of the vector lies along each of these axes. This is crucial because it allows us to work with vectors in a two-dimensional space more efficiently.

• Think of vector components as the building blocks of a vector.
• They help us understand the direction and magnitude of a vector.

For example, in the vector \(-2i + j\), the \(-2i\) represents a movement to the left (negative direction on the x-axis), while \(j\) represents an upward movement (positive direction on the y-axis). Breaking down vectors into components simplifies operations like addition and subtraction.

Horizontal Component

The horizontal component measures how far left or right a vector moves, paralleling the x-axis. It is represented with the letter \(i\), which corresponds to a unit vector in the horizontal direction. To determine the horizontal component of a vector like \(-2i + j\), focus on the value that multiplies \(i\). In this case, it's \(-2\), indicating a leftward movement since it's negative.
When adding vectors, we only add their respective components. This principle simplifies vector addition drastically.

• Horizontal component addition is independent of the vertical component.
• Considers only the directional influence along the x-axis.

In our example, when adding vectors \(-2i+j\) and \(2i-4j\), the horizontal components are \(-2\) and \(2\). Their sum is zero, meaning the resultant vector has no horizontal movement.

Vertical Component

The vertical component is similar to the horizontal one but occurs along the y-axis. It is denoted by \(j\) and shows how much a vector moves upward or downward. For our vector \(-2i + j\), the vertical component is \(1\), suggesting an upward movement.
Understanding the vertical component helps us determine how much a vector 'climbs' or 'descends' in a planar space. When adding vertical components from two vectors, the resulting value combines their upward and downward influences.
Taking the vectors \(-2i + j\) and \(2i - 4j\), the vertical components, \(1\) and \(-4\), add up to \(-3\). This means the combined effect of these vectors results in a net downward movement.

Resultant Vector

A resultant vector is what you get after combining several vectors—essentially, it’s the final vector that accounts for all vector components at once. To find it, you'll add both the horizontal and vertical components separately.

• Add all horizontal components together to get the resultant's horizontal part.
• Add all vertical components for the resultant's vertical part.

In our exercise, after combining the horizontal components (which gave us zero) and vertical components (which gave us \(-3\)), the resultant vector is \(0i - 3j\). This resultant vector represents the total effect of the original vectors in the horizontal and vertical directions combined, making it very useful in determining the total displacement or force direction in physics problems.