Problem 22 - Algebra and Trigonometry

Question

19-22 Find the component of \(\mathbf{u}\) along \(\mathbf{v}\). $$\mathbf{u}=7 \mathbf{i}, \quad \mathbf{v}=8 \mathbf{i}+6 \mathbf{j}$$

Step-by-Step Solution

Answer

The component of \( \mathbf{u} \) along \( \mathbf{v} \) is \( 4.48 \mathbf{i} + 3.36 \mathbf{j} \).

Step 1: Understand the Problem

We need to find the component of vector \( \mathbf{u} \) along vector \( \mathbf{v} \). This involves projecting \( \mathbf{u} \) onto \( \mathbf{v} \). The formula for the projection of \( \mathbf{u} \) onto \( \mathbf{v} \) is \( \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \).

Step 2: Calculate the Dot Product \(\mathbf{u} \cdot \mathbf{v}\)

Calculate the dot product of \( \mathbf{u} = 7 \mathbf{i} \) and \( \mathbf{v} = 8 \mathbf{i} + 6 \mathbf{j} \). The dot product formula is \( \mathbf{u} \cdot \mathbf{v} = u_1v_1 + u_2v_2 \). Here, \( u_2 = 0 \) and \( v_2 = 6 \), so:\[ 7 \mathbf{i} \cdot (8 \mathbf{i} + 6 \mathbf{j}) = 7 \times 8 + 0 \times 6 = 56 \].

Step 3: Calculate \(\mathbf{v} \cdot \mathbf{v}\)

The next step is to find \( \mathbf{v} \cdot \mathbf{v} \), which is \( (8 \mathbf{i} + 6 \mathbf{j}) \cdot (8 \mathbf{i} + 6 \mathbf{j}) \). Calculate using the dot product formula:\[ 8 \times 8 + 6 \times 6 = 64 + 36 = 100 \].

Step 4: Calculate the Projection

Use the projection formula \( \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \) with the values found:\[ \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{56}{100} (8 \mathbf{i} + 6 \mathbf{j}) = 0.56 (8 \mathbf{i} + 6 \mathbf{j}) \].

Step 5: Simplify the Result

Multiply the scalar \( 0.56 \) with the components of \( \mathbf{v} \):\[ 0.56 \times 8 \mathbf{i} = 4.48 \mathbf{i} \]\[ 0.56 \times 6 \mathbf{j} = 3.36 \mathbf{j} \]Thus, \( \text{proj}_{\mathbf{v}} \mathbf{u} = 4.48 \mathbf{i} + 3.36 \mathbf{j} \).

Key Concepts

Understanding Dot Product

The dot product is a mathematical operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. It's a crucial concept when dealing with vectors, particularly in physics and engineering. To calculate the dot product of two vectors, you multiply corresponding components and then sum these products.

For vectors \( \mathbf{u} = u_1 \mathbf{i} + u_2 \mathbf{j} \) and \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} \), the dot product is \( \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 \).

• \( u_1 \cdot v_1 \) involves the multiplication of their respective \(i\) component.
• \( u_2 \cdot v_2 \) handles the multiplication of their respective \(j\) component.

In the example we have: \( \mathbf{u} = 7 \mathbf{i} \) and \( \mathbf{v} = 8 \mathbf{i} + 6 \mathbf{j} \). By substituting these into the dot product formula: \( 7 \cdot 8 + 0 \cdot 6 = 56 \). Note that although \( \mathbf{u} \) doesn't have a \(j\) component, we still include a \(0\) for \(u_2\) to complete the operation.

Exploring Vector Components

Vector components break down a vector into its constituent parts. Each vector in a two-dimensional space has two components, often represented by \( \mathbf{i} \) and \( \mathbf{j} \), corresponding to the x and y axes. These components help in performing calculations like vector addition, subtraction, and projections.

For instance, the vector \( \mathbf{v} = 8 \mathbf{i} + 6 \mathbf{j} \) has two components:

• The \( \mathbf{i} \) component, which is parallel to the x-axis and has a magnitude of 8.
• The \( \mathbf{j} \) component, which points along the y-axis with a magnitude of 6.

Understanding these components is essential when calculating the length (or magnitude) of a vector or projecting another vector onto it. When projecting, these components are scaled and directed to show shadow or line effects of one vector on another. It’s like a breakdown of a vector’s influence in each direction of a given space.

Using the Projection Formula

The projection formula helps to find how much of one vector lies along another vector. This is particularly useful in physics for resolving forces and in computer graphics for shading effects. The formula for projecting vector \( \mathbf{u} \) onto vector \( \mathbf{v} \) is given by:The formula:\[ \text{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \]Derived by calculating how much influence \( \mathbf{u} \) has along \( \mathbf{v} \). This ratio is usually a scalar that scales \( \mathbf{v} \) to a form aligned with \( \mathbf{u} \).

In our exercise, calculating this involves several steps:

• Find the dot product \( \mathbf{u} \cdot \mathbf{v} = 56 \).
• Compute \( \mathbf{v} \cdot \mathbf{v} = 100 \).
• Use the projection formula \( \frac{56}{100} (8 \mathbf{i} + 6 \mathbf{j}) = 0.56 (8 \mathbf{i} + 6 \mathbf{j}) \).
• Simplify to \( 4.48 \mathbf{i} + 3.36 \mathbf{j} \).

This step results in a new vector showing how much of \( \mathbf{u} \) falls along \( \mathbf{v} \), creating a parallel and scaled version of \( \mathbf{v} \). This makes the concept of projection both practical and visually intuitive.