Problem 27 - 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 \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}\). $$ \operatorname{proj}_{\mathbf{j}} \mathbf{u} $$

Step-by-Step Solution

Answer

\( \operatorname{proj}_{\mathbf{j}} \mathbf{u} = 2\mathbf{j} \)

Step 1: Identify Components

The vector \( \mathbf{u} \) is given as \( \mathbf{u} = \mathbf{i} + 2 \mathbf{j} \). Here, the coefficient of \( \mathbf{j} \) in \( \mathbf{u} \) is 2.

Step 2: Apply Projection Formula

The projection of \( \mathbf{u} \) onto \( \mathbf{j} \) using the formula for projection is given by \( \operatorname{proj}_{\mathbf{j}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{j}}{\mathbf{j} \cdot \mathbf{j}} \mathbf{j} \).

Step 3: Calculate Dot Products

First, calculate \( \mathbf{u} \cdot \mathbf{j} = (\mathbf{i} + 2\mathbf{j}) \cdot (0\mathbf{i} + 1\mathbf{j}) = 0 \cdot 1 + 2 \cdot 1 = 2 \). Next, \( \mathbf{j} \cdot \mathbf{j} = (0\mathbf{i} + 1\mathbf{j}) \cdot (0\mathbf{i} + 1\mathbf{j}) = 0 \cdot 0 + 1 \cdot 1 = 1 \).

Step 4: Calculate the Projection

Substitute these dot products into the projection formula: \( \operatorname{proj}_{\mathbf{j}} \mathbf{u} = \frac{2}{1} \mathbf{j} = 2\mathbf{j} \).

Key Concepts

Dot Product

The dot product is a fundamental operation used in vector mathematics. It helps us understand how two vectors relate to each other through projection.
In simple terms, the dot product of two vectors gives us a single number that represents how much one vector extends in the direction of another.
To compute the dot product, we usually use the formula:

• If two vectors are given as \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), their dot product is \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).

Understanding the dot product is crucial because it serves as the basis for calculating the projection of one vector onto another. In the example given, the dot product of \( \mathbf{u} \) and \( \mathbf{j} \) was calculated as 2. This number is critical for determining how \( \mathbf{u} \) sits in relation to \( \mathbf{j} \).

Projection Formula

The projection formula allows us to project one vector onto another, providing us with a vector that is in the direction of the given vector.
To project a vector \( \mathbf{a} \) onto another vector \( \mathbf{b} \), we use the formula:

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

This formula tells us how much of \( \mathbf{a} \) is in the direction of \( \mathbf{b} \).
Consider it as projecting a shadow of \( \mathbf{a} \) onto \( \mathbf{b} \). For the given problem, the projection of \( \mathbf{u} \) onto \( \mathbf{j} \) results in \( 2\mathbf{j} \). This means the part of \( \mathbf{u} \) that points in the direction of \( \mathbf{j} \) is exactly \( 2\mathbf{j} \).
The projection lets us express one vector in relation to another in a directed and scalar way.

Vector Components

Vector components are what make up a vector in any system.
We consider these components to understand and express the vector fully.
For instance, in the two-dimensional plane with basis vectors \( \mathbf{i} \) and \( \mathbf{j} \), a vector \( \mathbf{a} \) can be expressed as \( a_1\mathbf{i} + a_2\mathbf{j} \), where \( a_1 \) and \( a_2 \) are its components.

• The first component (like \( a_1 \)) reflects the extent of the vector along the horizontal axis (\( \mathbf{i} \)).
• The second component (like \( a_2 \)) reflects how far the vector moves along the vertical axis (\( \mathbf{j} \)).

In our case, \( \mathbf{u} \) is described by its components, where it moves 1 unit in the \( \mathbf{i} \) direction and 2 units in the \( \mathbf{j} \) direction.
This description is essential when calculating projections.
Knowing these components allows us to work out projections and understand the vector in terms of real-world applications easily.