Problem 44 - Advanced Engineering Mathematics

Question

In Problems 43 and 44, \(\mathbf{a}=4 \mathbf{i}+3 \mathbf{j}\) and \(\mathbf{b}=-\mathbf{i}+\mathbf{j}\). Find the indicated vector. proj \(_{(\mathbf{a}-b)} \mathbf{b}\)

Step-by-Step Solution

Answer

The projection is \(-\frac{15}{29}\mathbf{i} - \frac{6}{29}\mathbf{j}\).

Step 1: Understand the Projection Formula

The projection of vector \(\mathbf{b}\) onto vector \(\mathbf{v}\), denoted as \(\text{proj}_{\mathbf{v}} \mathbf{b}\), is given by the formula: \[ \text{proj}_{\mathbf{v}} \mathbf{b} = \frac{\mathbf{b} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \] where \(\mathbf{b} \cdot \mathbf{v}\) is the dot product of vectors \(\mathbf{b}\) and \(\mathbf{v}\), and \(\mathbf{v} \cdot \mathbf{v}\) is the dot product of \(\mathbf{v}\) with itself.

Step 2: Find the Vector \(\mathbf{a} - \mathbf{b}\)

First, calculate the vector \(\mathbf{a} - \mathbf{b}\):\[ \mathbf{a} = 4\mathbf{i} + 3\mathbf{j} \]\[ \mathbf{b} = -\mathbf{i} + \mathbf{j} \]\[ \mathbf{a} - \mathbf{b} = (4\mathbf{i} + 3\mathbf{j}) - (-\mathbf{i} + \mathbf{j}) = 5\mathbf{i} + 2\mathbf{j} \]

Step 3: Calculate the Dot Product \(\mathbf{b} \cdot (\mathbf{a} - \mathbf{b})\)

The dot product \(\mathbf{b} \cdot (\mathbf{a} - \mathbf{b})\) is calculated as:\[ \mathbf{b} = -\mathbf{i} + \mathbf{j} \]\[ \mathbf{a} - \mathbf{b} = 5\mathbf{i} + 2\mathbf{j} \]\[ \mathbf{b} \cdot (\mathbf{a} - \mathbf{b}) = (-1)(5) + (1)(2) = -5 + 2 = -3 \]

Step 4: Calculate the Dot Product \((\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b})\)

Calculate the dot product of \(\mathbf{a} - \mathbf{b}\) with itself:\[ (\mathbf{a} - \mathbf{b}) \cdot (\mathbf{a} - \mathbf{b}) = (5\mathbf{i} + 2\mathbf{j}) \cdot (5\mathbf{i} + 2\mathbf{j}) = 5^2 + 2^2 = 25 + 4 = 29 \]

Step 5: Compute the Projection \(\text{proj}_{(\mathbf{a} - \mathbf{b})} \mathbf{b}\)

Now plug in the values into the projection formula:\[ \text{proj}_{(\mathbf{a} - \mathbf{b})} \mathbf{b} = \frac{-3}{29} (5\mathbf{i} + 2\mathbf{j}) \]\[ = \left(-\frac{15}{29}\right)\mathbf{i} + \left(-\frac{6}{29}\right)\mathbf{j} \]

Step 6: Write the Final Result

The vector projection of \(\mathbf{b}\) onto \(\mathbf{a} - \mathbf{b}\) is:\[ \text{proj}_{(\mathbf{a} - \mathbf{b})} \mathbf{b} = -\frac{15}{29}\mathbf{i} - \frac{6}{29}\mathbf{j} \]

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation in linear algebra. It combines two vectors to produce a single number. This operation is crucial because it helps determine the angle between vectors and is key in finding projections. To compute the dot product of two vectors \( \mathbf{u} = u_1\mathbf{i} + u_2\mathbf{j} \) and \( \mathbf{v} = v_1\mathbf{i} + v_2\mathbf{j} \), you multiply their corresponding components and add the results. The formula is:\[ \mathbf{u} \cdot \mathbf{v} = u_1 v_1 + u_2 v_2 \]

• The result is a scalar, a single numerical value.
• This product gives zero if the vectors are perpendicular.
• It's essential for formulas involving angles and projections.

Understanding the concept of dot product allows us to effectively work with vectors' orientations and magnitudes, helping solve complex geometric problems in vector calculus and physics.

Vector Subtraction

Vector subtraction is an operation that allows us to compute the difference between two vectors. When we subtract one vector from another, we effectively find a vector that "bridges" the tip of the first vector to the tip of the second. For vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \), the subtraction \( \mathbf{a} - \mathbf{b} \) is calculated as:\[ \mathbf{a} - \mathbf{b} = (a_1 - b_1)\mathbf{i} + (a_2 - b_2)\mathbf{j} \]

• Each component of the result vector is the difference of the corresponding components of \( \mathbf{a} \) and \( \mathbf{b} \).
• This operation can change both the direction and length of the original vectors.
• Vector subtraction is often used to find relative movement or position differences.

Knowing how to subtract vectors plays a significant role in solving navigation problems, computer graphics, and more, where paths and positions need to be dynamically calculated.

Projection Formula

Projecting one vector onto another involves finding a parallel component along the second vector. The projection formula is crucial when projecting vector \( \mathbf{b} \) onto a vector \( \mathbf{v} \). The formula is:\[ \text{proj}_{\mathbf{v}} \mathbf{b} = \frac{\mathbf{b} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v} \]

• The numerator \( \mathbf{b} \cdot \mathbf{v} \) checks alignment by the dot product.
• The denominator \( \mathbf{v} \cdot \mathbf{v} \) normalizes for \( \mathbf{v} \)'s length.
• Gives the component of \( \mathbf{b} \) that points along \( \mathbf{v} \).

This formula is widely used in physics to resolve forces or velocities into components that align with different directions, such as inclined planes, optimizing processes such as lighting, and rendering in computer graphics.

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, matrices, and vector spaces. It's fundamental in various scientific areas, including engineering, computer science, physics, and more. Key ideas include operations involving vectors and matrices, solving systems of linear equations, and understanding transformations. - **Vectors and Scalars**: Vectors have magnitude and direction, while scalars are single values. - **Vector Spaces**: Collections of vectors where addition and scalar multiplication are defined. - **Matrices**: Rectangular arrays of numbers that represent linear transformations between vector spaces. Applications of linear algebra are vast. They include machine learning algorithms, 3D modeling in graphics engines, and solving differential equations. Linear algebra tools let us handle large systems of equations or perform operations on data more efficiently. Whether projecting vectors or analyzing multidimensional data, linear algebra provides the theoretical foundation and practical tools necessary for advanced computation and analysis.