Problem 31 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find each of the given projections if \(\mathbf{u}=3 \mathbf{i}+2 \mathbf{j}+\mathbf{k}, \mathbf{v}=2 \mathbf{i}-\mathbf{k}\), and \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). \(\operatorname{proj}_{\mathbf{u}} \mathbf{w}\)

Step-by-Step Solution

Answer

\( \operatorname{proj}_{\mathbf{u}} \mathbf{w} = \frac{15}{7} \mathbf{i} + \frac{10}{7} \mathbf{j} + \frac{5}{7} \mathbf{k} \).

Step 1: Understand the Formula for Projection

The projection of a vector \( \mathbf{a} \) onto \( \mathbf{b} \) is given by the formula \( \operatorname{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b} \). In this problem, we will project \( \mathbf{w} \) onto \( \mathbf{u} \).

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

The dot product \( \mathbf{w} \cdot \mathbf{u} \) is calculated as follows: \[ \mathbf{w} \cdot \mathbf{u} = (1)(3) + (5)(2) + (-3)(1) = 3 + 10 - 3 = 10 \].

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

The dot product \( \mathbf{u} \cdot \mathbf{u} \) is calculated by \[ \mathbf{u} \cdot \mathbf{u} = (3)(3) + (2)(2) + (1)(1) = 9 + 4 + 1 = 14 \].

Step 4: Plug into the Projection Formula

Now plug the results into the projection formula: \[ \operatorname{proj}_{\mathbf{u}} \mathbf{w} = \frac{10}{14} \mathbf{u} \] which simplifies to \[ \operatorname{proj}_{\mathbf{u}} \mathbf{w} = \frac{5}{7} \mathbf{u} \].

Step 5: Calculate the Projected Vector

Now, calculate \( \frac{5}{7} \mathbf{u} \):\[ \operatorname{proj}_{\mathbf{u}} \mathbf{w} = \frac{5}{7} (3 \mathbf{i} + 2 \mathbf{j} + \mathbf{k}) \] \[ = \frac{15}{7} \mathbf{i} + \frac{10}{7} \mathbf{j} + \frac{5}{7} \mathbf{k} \].

Key Concepts

Dot Product

The dot product is a fundamental concept in vector calculus that helps us discover interactions between vectors by producing a single number, often called a scalar. To find the dot product of two vectors, you multiply their corresponding components and then sum these products. For instance, consider the vectors \( \mathbf{u} = 3 \mathbf{i} + 2 \mathbf{j} + \mathbf{k} \) and \( \mathbf{v} = 2 \mathbf{i} - \mathbf{k} \). The dot product \( \mathbf{u} \cdot \mathbf{v} \) would be computed as:

• Multiply the \( \mathbf{i} \)-components: \( 3 \times 2 = 6 \)
• Multiply the \( \mathbf{j} \)-components: \( 2 \times 0 = 0 \) (since \( \mathbf{v} \) has no \( \mathbf{j} \) component)
• Multiply the \( \mathbf{k} \)-components: \( 1 \times (-1) = -1 \)

Add these values: \( 6 + 0 - 1 = 5 \).
Thus, the dot product \( \mathbf{u} \cdot \mathbf{v} = 5 \). This simple calculation helps in determining the projection of one vector onto another, which we will explore further in the next sections.

Vector Calculus

Vector calculus is a branch of mathematics that deals with vector fields and differential operators, expanding on the mathematical operations you can perform using vectors.
It allows us to describe physical quantities such as velocity and force across fields, offering tools like gradient, divergence, and curl for manipulating vector fields. In vector calculus, calculation involves not just simple addition or subtraction of vector quantities but also more complex operations like cross products, dot products, and projections.
These operations help in understanding the motion of particles through space, electric and magnetic fields, and even fluid dynamics. Working with vectors often requires combining them to understand their alignment or orthogonality. The dot product helps determine if two vectors are perpendicular (orthogonal), as a dot product of zero indicates this. When working on projections, these vectors' interactions in spatial contexts become clearer through vector calculus principles.

Projection Formula

The projection formula is a powerful tool in vector calculus, allowing us to "drop" one vector onto another to see how much of the first vector points in the direction of the second. The formula for the projection of a vector \( \mathbf{a} \) onto \( \mathbf{b} \) is provided by:\[ \operatorname{proj}_{\mathbf{b}} \mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b} \]This formula not only helps in visualizing how vectors interact but also comes in handy in practical applications such as computer graphics and physics.To apply this, we calculate:

• The dot product \( \mathbf{a} \cdot \mathbf{b} \), giving a scalar measuring the length component of \( \mathbf{a} \) projected onto \( \mathbf{b} \).
• The dot product \( \mathbf{b} \cdot \mathbf{b} \), the magnitude squared of \( \mathbf{b} \).
• Multiply the result by the vector \( \mathbf{b} \) itself, scaling \( \mathbf{b} \) by the projection's magnitude.

By using this method, manipulating directional components of vectors becomes straightforward, as seen in step-wise solutions of projection problems, including projecting \( \mathbf{w} \) onto \( \mathbf{u} \). This dimensional breakdown transforms abstract vector ideas into tangible, spatial concepts.