Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). \(\mathrm{pro} \mathrm{j}_{\mathrm{j}} \mathbf{u}\)

Step-by-Step Solution

Answer

The projection is \( 2\mathbf{j} \).

Step 1: Recognize the Problem

We need to find the projection of vector \( \mathbf{u} \) onto vector \( \mathbf{j} \). This involves using the formula for vector projection.

Step 2: Write the Projection Formula

The formula for the projection of a vector \( \mathbf{a} \) onto \( \mathbf{b} \) is \( \mathrm{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b} \).

Step 3: Identify Vectors for Calculation

Here, \( \mathbf{u} = \mathbf{i} + 2 \mathbf{j} \) and \( \mathbf{j} = 0\mathbf{i} + 1\mathbf{j} \). We need to substitute these vectors into the projection formula.

Step 4: Compute the Dot Product \( \mathbf{u} \cdot \mathbf{j} \)

The dot product is computed as \( (\mathbf{i} + 2 \mathbf{j}) \cdot (0\mathbf{i} + 1\mathbf{j}) = 1(0) + 2(1) = 2 \).

Step 5: Compute the Dot Product \( \mathbf{j} \cdot \mathbf{j} \)

Since \( \mathbf{j} = 0\mathbf{i} + 1\mathbf{j} \), the dot product is \( (0\mathbf{i} + 1\mathbf{j}) \cdot (0\mathbf{i} + 1\mathbf{j}) = 0^2 + 1^2 = 1 \).

Step 6: Apply the Projection Formula

Using the results from Steps 4 and 5, we have: \[ \mathrm{proj}_{\mathbf{j}} \mathbf{u} = \frac{2}{1} (0\mathbf{i} + 1\mathbf{j}) = 2\mathbf{j} \].

Step 7: Conclusion

Thus, the projection of \( \mathbf{u} \) onto \( \mathbf{j} \) is \( 2\mathbf{j} \).

Key Concepts

Understanding the Dot Product

When working with vectors, the dot product is a crucial operation used in various calculations, such as finding vector projections. The dot product of two vectors results in a scalar. If you have two vectors, \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), their dot product is calculated as:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).

This operation multiplies the corresponding components of the vectors and sums them together. In the given problem, understanding how to calculate the dot product allows us to move forward with applying other vector operations, such as the projection formula.

Exploring the Projection Formula

The projection formula is used to project one vector onto another. The result of a projection is a vector. The formula for projecting vector \( \mathbf{a} \) onto vector \( \mathbf{b} \) is given by:

• \( \mathrm{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b} \)

To understand this formula, let's break it down:

• First, you calculate the dot product of the vectors \( \mathbf{a} \) and \( \mathbf{b} \). This gives a scalar that tells new information on the magnitudes of those vectors.
• Then, you calculate the dot product of \( \mathbf{b} \) with itself to normalize the vector \( \mathbf{b} \). This ensures that the projection is correctly scaled.
• Finally, multiply the normalized vector \( \mathbf{b} \) by the scalar obtained. The outcome is the projection vector.

Introduction to Vectors

Vectors are fundamental in calculus and physics, representing quantities that have both magnitude and direction. In plane space (2D), we often use the basis vectors \( \mathbf{i} \) and \( \mathbf{j} \) to express vectors. For example, \( \mathbf{u} = \mathbf{i} + 2\mathbf{j} \) is a vector with a horizontal component of 1 and a vertical component of 2.
Vectors facilitate the solving of various geometric and physical problems by allowing complex multi-dimensional data to be represented simply.

• Vector Notation: Expressed as combinations of \( \mathbf{i} \) and \( \mathbf{j} \).
• Magnitude of a Vector: Using Pythagorean theorem for any vector \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \), \( \|\mathbf{a}\| = \sqrt{a_1^2 + a_2^2} \).
• Applications: Frequently used for indicating points, directions and transformations in space.

Problem Solving in Calculus with Vectors

Vector calculus brings a unique perspective to calculus problem solving. It allows us to explore calculus concepts such as derivatives and integrals in multi-dimensional space, making it especially useful in physics and engineering. When working with vector problems, it's vital to:

• Understand vector operations:
• Be proficient in dot product, cross product, and projections.
• Set up problems accurately:
• Draw diagrams to visualize vectors and their interactions.
• Assign coordinates to vectors when necessary.
• Apply the right formulas:
• Use projection formulas to break down problems into smaller parts.
Identifying patterns in vector problems can often reveal simpler approaches to otherwise complex solutions. Whether for solving projections or more intricate integrations involving vectors, the methodology remains crucial in achieving accurate results.