Problem 32 - Precalculus with Calculus Previews

Question

If \(\mathbf{u}=3 \mathbf{i}-\mathbf{j}\) and \(\mathbf{v}=2 \mathbf{i}+4 \mathbf{j},\) find the horizontal and the vertical components of the indicated vector. $$ 4(\mathbf{u}+3 \mathbf{v}) $$

Step-by-Step Solution

Answer

Horizontal: 36, Vertical: 44.

Step 1: Calculate 3\mathbf{v}

To find \(3 \mathbf{v}\), multiply each component of \(\mathbf{v} = 2 \mathbf{i} + 4 \mathbf{j}\) by 3:- Horizontal component: \(3 \times 2\mathbf{i} = 6\mathbf{i}\)- Vertical component: \(3 \times 4\mathbf{j} = 12\mathbf{j}\)Thus, \(3 \mathbf{v} = 6\mathbf{i} + 12\mathbf{j}\).

Step 2: Add \mathbf{u} and 3\mathbf{v}

Now add \(\mathbf{u} = 3 \mathbf{i} - \mathbf{j}\) and \(3 \mathbf{v} = 6 \mathbf{i} + 12 \mathbf{j}\) together:- Horizontal component: \(3\mathbf{i} + 6\mathbf{i} = 9\mathbf{i}\) - Vertical component: \(-\mathbf{j} + 12\mathbf{j} = 11\mathbf{j}\)Thus, \(\mathbf{u} + 3\mathbf{v} = 9\mathbf{i} + 11\mathbf{j}\).

Step 3: Calculate 4(\mathbf{u}+3\mathbf{v})

To find the vector \(4(\mathbf{u}+3\mathbf{v})\), multiply each component of \(\mathbf{u} + 3\mathbf{v} = 9 \mathbf{i} + 11 \mathbf{j}\) by 4:- Horizontal component: \(4 \times 9\mathbf{i} = 36\mathbf{i}\)- Vertical component: \(4 \times 11\mathbf{j} = 44\mathbf{j}\)Thus, the vector is \(4(\mathbf{u} + 3\mathbf{v}) = 36\mathbf{i} + 44\mathbf{j}\).

Step 4: Identify Horizontal and Vertical Components

The components you have calculated can now be directly identified as the horizontal and vertical components:- The horizontal component is \(36\).- The vertical component is \(44\).

Key Concepts

Vector Addition

When we talk about vector addition, we usually refer to the process of combining two or more vectors to produce a resultant vector. In mathematics, vectors are represented as arrows that have both magnitude (length) and direction. To add vectors, each component of the vectors must be added separately.
For example, let's say we have two vectors: \( \mathbf{u} = 3 \mathbf{i} - \mathbf{j} \) and \( 3 \mathbf{v} = 6 \mathbf{i} + 12 \mathbf{j} \). When we add them, we include their horizontal and vertical components:

• The horizontal components: \( 3\mathbf{i} + 6\mathbf{i} = 9\mathbf{i} \)
• The vertical components: \( -\mathbf{j} + 12\mathbf{j} = 11\mathbf{j} \)

Thus, the sum \( \mathbf{u} + 3\mathbf{v} \) results in the vector \( 9\mathbf{i} + 11\mathbf{j} \), which combines both horizontal and vertical influences from the original vectors. This process is fundamental for understanding complex movements and forces acting in multiple directions.

Scalar Multiplication

Scalar multiplication involves multiplying a vector by a scalar (real number). This operation affects the magnitude of the vector without changing its direction, except in cases where the scalar is negative, which flips the vector direction.
Take the vector \( \mathbf{v} = 2 \mathbf{i} + 4 \mathbf{j} \) as an example. To compute \( 3 \mathbf{v} \), we multiply each component by 3:

• Horizontal component: \( 3 \times 2 \mathbf{i} = 6 \mathbf{i} \)
• Vertical component: \( 3 \times 4 \mathbf{j} = 12 \mathbf{j} \)

This results in the vector \( 6 \mathbf{i} + 12 \mathbf{j} \). Similarly, further scaling the resultant vector \( 9 \mathbf{i} + 11 \mathbf{j} \) by 4 yields:

• Horizontal component: \( 4 \times 9 \mathbf{i} = 36 \mathbf{i} \)
• Vertical component: \( 4 \times 11 \mathbf{j} = 44 \mathbf{j} \)

Ultimately, scalar multiplication is a powerful tool in vector manipulation for both enlarging and reducing vector magnitudes according to different needs.

Horizontal Component

The horizontal component of a vector represents how far the vector extends along the horizontal axis. It is crucial in breaking down a vector into a more manageable form, especially when dealing with two-dimensional problems.
For instance, when determining the horizontal component of the vector \( 4(\mathbf{u}+3\mathbf{v}) \), you multiply the horizontal part of \( \mathbf{u}+3\mathbf{v} = 9 \mathbf{i} + 11 \mathbf{j} \) by 4:

• Horizontal: \( 4 \times 9 \mathbf{i} = 36 \mathbf{i} \)

This gives us the complete horizontal influence the original vectors have after addition and scalar multiplication. Knowing the horizontal component allows for a clearer understanding of movements or forces acting in the horizontal direction of whatever context the vectors are applied. Hence, the horizontal component simplifies calculations and assists in accurately modeling physical phenomena.

Vertical Component

The vertical component corresponds to how much a vector stretches or affects the vertical axis. Distinguishing the vertical component is essential in physics and engineering to understand vertical forces and motions.
In our example of vector operations, we calculated the vertical component after various operations: \( \mathbf{u}+3\mathbf{v} = 9 \mathbf{i} + 11 \mathbf{j} \). To find the extended vertical component for \( 4(\mathbf{u}+3\mathbf{v}) \), you perform scalar multiplication on the vertical part:

• Vertical: \( 4 \times 11 \mathbf{j} = 44 \mathbf{j} \)

The vertical component here is 44, showing how the combined operations on the initial vectors influence the vertical direction. Isolating the vertical component in calculations is crucial for accurately predicting outcomes when vectors impact systems, helping engineers and scientists design and analyze efficient solutions.