Problem 29 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 29-34, 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 \(\mathbf{w}=\mathbf{i}+5 \mathbf{j}-3 \mathbf{k}\). $$ \operatorname{proj}_{\mathbf{v}} \mathbf{u} $$

Step-by-Step Solution

Answer

The projection of \( \mathbf{u} \) onto \( \mathbf{v} \) is \( 2\mathbf{i} - \mathbf{k} \).

Step 1: Understand the Projection Formula

The projection of a vector \( \mathbf{u} \) onto another vector \( \mathbf{v} \) is given by the formula: \[ \operatorname{proj}_{\mathbf{v}} \mathbf{u} = \left( \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \right) \mathbf{v} \] where \( \cdot \) denotes the dot product.

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

To compute \( \mathbf{u} \cdot \mathbf{v} \), multiply corresponding components of \( \mathbf{u} \) and \( \mathbf{v} \) and sum them: \[ \mathbf{u} \cdot \mathbf{v} = (3)(2) + (2)(0) + (1)(-1) = 6 + 0 - 1 = 5 \]

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

To find \( \mathbf{v} \cdot \mathbf{v} \), multiply corresponding components of \( \mathbf{v} \) and sum: \[ \mathbf{v} \cdot \mathbf{v} = (2)(2) + (0)(0) + (-1)(-1) = 4 + 0 + 1 = 5 \]

Step 4: Compute the Projection Scalar

Using the results from Steps 2 and 3, determine the scalar for the projection: \[ \frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} = \frac{5}{5} = 1 \]

Step 5: Calculate the Projection Vector

Multiply the scalar from Step 4 by vector \( \mathbf{v} \) to find the projection: \[ \operatorname{proj}_{\mathbf{v}} \mathbf{u} = 1 \times (2\mathbf{i} - \mathbf{k}) = 2\mathbf{i} - \mathbf{k} \]

Key Concepts

Dot Product

The dot product is a fundamental operation in linear algebra and vector mathematics. It helps us measure how much of one vector points in the direction of another. The dot product of two vectors, \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k} \), is calculated as follows:\[\mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 + a_3 b_3\]This operation returns a scalar, indicating the magnitude of the vectors' alignment.

• If the dot product is zero, the vectors are perpendicular.
• If it is positive, they point in generally the same direction.
• If it's negative, they point in opposite directions.

Understanding and applying the dot product is essential for operations such as finding vector projections and calculating angles between vectors. This is because it captures how much one vector aligns with another, providing insight into geometrical relationships in space.
In our given problem, the dot product \( \mathbf{u} \cdot \mathbf{v} \) was calculated to be 5.

Vector Operations

Vector operations include a variety of mathematical procedures, such as addition, subtraction, and scalar multiplication. These operations allow us to manipulate vectors in a multitude of ways.**Basic Operations:*** **Addition and Subtraction** of vectors are carried out component-wise. For instance, for vectors \( \mathbf{a} \) and \( \mathbf{b} \), the sum \( \mathbf{a} + \mathbf{b} \) would be calculated as: \[ (a_1 + b_1)\mathbf{i} + (a_2 + b_2)\mathbf{j} + (a_3 + b_3)\mathbf{k} \]* **Scalar Multiplication** involves multiplying each component of the vector by a scalar value: \[ c \times \mathbf{a} = (ca_1)\mathbf{i} + (ca_2)\mathbf{j} + (ca_3)\mathbf{k} \]These operations are crucial when working with vector projections. In the original exercise, scalar multiplication was used to scale the vector \( \mathbf{v} \) after finding the projection scalar. This resulted in the projection vector.
Understanding these operations helps in visualizing and solving more complex vector problems, such as those found in physics and engineering scenarios.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It provides a framework for understanding more advanced concepts such as transformations, projections, and eigenvalues.**Key Concepts in Linear Algebra:**
* **Vector Spaces:** Fundamental structures that allow vectors to be added together and multiplied by scalars.* **Matrices:** Arrays of numbers that represent linear transformations. They simplify systems of linear equations.* **Projections:** A way of depicting vectors onto a line or plane. This concept makes extensive use of the dot product.Projections within linear algebra are particularly useful because they help us understand how vectors interact within a space. For example, projecting \( \mathbf{u} \) onto \( \mathbf{v} \) shows which component of \( \mathbf{u} \) runs parallel to \( \mathbf{v} \).
In our original exercise, the formula for projection, which uses both the dot product and scalar multiplication, is a quintessential application of linear algebra principles. These principles are not only theoretical but also applicable in computer graphics, optimization problems, and many scientific computations.