Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 23-28, find each of the given projections if \(\mathbf{u}=\mathbf{i}+2 \mathbf{j}, \mathbf{v}=2 \mathbf{i}-\mathbf{j}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). $$ \operatorname{proj}_{\mathbf{u}}(\mathbf{w}-\mathbf{v}) $$

Step-by-Step Solution

Answer

The projection is \( \frac{11}{5}\mathbf{i} + \frac{22}{5}\mathbf{j} \).

Step 1: Calculate \( \mathbf{w} - \mathbf{v} \)

Subtract vector \( \mathbf{v} = 2\mathbf{i} - \mathbf{j} \) from vector \( \mathbf{w} = \mathbf{i} + 5\mathbf{j} \). This results in:\[ \mathbf{w} - \mathbf{v} = (\mathbf{i} + 5\mathbf{j}) - (2\mathbf{i} - \mathbf{j}) = \mathbf{i} + 5\mathbf{j} - 2\mathbf{i} + \mathbf{j} = -\mathbf{i} + 6\mathbf{j} \]

Step 2: Compute the Dot Product \( \mathbf{w} - \mathbf{v} \) and \( \mathbf{u} \)

Calculate the dot product of \( \mathbf{u} = \mathbf{i} + 2\mathbf{j} \) and \( \mathbf{w} - \mathbf{v} = -\mathbf{i} + 6\mathbf{j} \). This is calculated as:\[ \mathbf{u} \cdot (\mathbf{w} - \mathbf{v}) = (\mathbf{i} + 2\mathbf{j}) \cdot (-\mathbf{i} + 6\mathbf{j}) = (1)(-1) + (2)(6) = -1 + 12 = 11 \]

Step 3: Calculate the Dot Product \( \mathbf{u} \cdot \mathbf{u} \)

Find the dot product of \( \mathbf{u} \) with itself:\[ \mathbf{u} \cdot \mathbf{u} = (\mathbf{i} + 2\mathbf{j}) \cdot (\mathbf{i} + 2\mathbf{j}) = (1)(1) + (2)(2) = 1 + 4 = 5 \]

Step 4: Compute the Projection

Use the formula for the projection of a vector \( \mathbf{a} \) onto \( \mathbf{b} \), given by:\[ \operatorname{proj}_{\mathbf{u}}(\mathbf{w} - \mathbf{v}) = \frac{\mathbf{u} \cdot (\mathbf{w} - \mathbf{v})}{\mathbf{u} \cdot \mathbf{u}} \cdot \mathbf{u} \]Substitute the known values:\[ \operatorname{proj}_{\mathbf{u}}(\mathbf{w} - \mathbf{v}) = \frac{11}{5} \cdot (\mathbf{i} + 2\mathbf{j}) = \frac{11}{5}\mathbf{i} + \frac{22}{5}\mathbf{j} \]

Key Concepts

Dot Product

The dot product is a fundamental operation in vector mathematics. It combines two vectors and returns a scalar value. To compute the dot product between two vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), you multiply their corresponding components and then sum these products:

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

In the context of vector projections, the dot product is crucial because it helps to quantify how much of one vector is in the direction of another. It essentially measures their alignment. Thus, while solving for projections, you will repeatedly use the dot product to derive more details about how one vector "projects" onto another.

Vector Operations

Vector operations are the set of mathematical methods for manipulating vectors. Key operations include addition, subtraction, and scalar multiplication. In this exercise, we're particularly interested in subtraction.To subtract vectors, such as \( \mathbf{w} \) and \( \mathbf{v} \), you compute each component separately:

• \( \mathbf{w} - \mathbf{v} = (w_1 - v_1) \mathbf{i} + (w_2 - v_2) \mathbf{j} \)

So, for \( \mathbf{w} = \mathbf{i} + 5\mathbf{j} \) and \( \mathbf{v} = 2\mathbf{i} - \mathbf{j} \), the subtraction \( -\mathbf{i} + 6\mathbf{j} \) arises. This operation simplifies the problem, letting you focus on the relevant components in subsequent calculations. Understanding these basic operations helps solve more complex vector-related problems efficiently.

Projection Formula

The projection formula is a tool used to project one vector onto another. This involves determining how much of one vector goes in the direction of another. The mathematical expression for projecting vector \( \mathbf{a} \) onto \( \mathbf{b} \) is:\[\operatorname{proj}_{\mathbf{b}}(\mathbf{a}) = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \cdot \mathbf{b}\]

• The numerator \( \mathbf{a} \cdot \mathbf{b} \) computes the dot product of the vectors, capturing their directional alignment.
• The denominator \( \mathbf{b} \cdot \mathbf{b} \) finds the magnitude squared of \( \mathbf{b} \), ensuring the division scales the projection correctly.

This formula outputs a new vector directed along \( \mathbf{b} \), representing how much of \( \mathbf{a} \) aligns with \( \mathbf{b} \). In our problem, we used this formula to find \( \text{proj}_{\mathbf{u}}(\mathbf{w} - \mathbf{v}) \), showing how \( \mathbf{w} - \mathbf{v} \) projects onto \( \mathbf{u} \). The result helps in understanding the relationship between these vectors in terms of direction and magnitude.