Problem 23 - Precalculus Mathematics for Calculus

Question

Find the indicated quantity, assuming \(\mathbf{u}=2 \mathbf{i}+\mathbf{j}, \mathbf{v}=\mathbf{i}-3 \mathbf{j},\) and \(\mathbf{w}=3 \mathbf{i}+4 \mathbf{j}\). $$(\mathbf{u}+\mathbf{v}) \cdot(\mathbf{u}-\mathbf{v})$$

Step-by-Step Solution

Answer

The dot product is -5.

Step 1: Addition of Vectors

First, find \( \mathbf{u} + \mathbf{v} \).Given \( \mathbf{u} = 2\mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = \mathbf{i} - 3\mathbf{j} \), the addition is:\[ \mathbf{u} + \mathbf{v} = (2\mathbf{i} + \mathbf{j}) + (\mathbf{i} - 3\mathbf{j}) = (2 + 1)\mathbf{i} + (1 - 3)\mathbf{j} = 3\mathbf{i} - 2\mathbf{j}. \]

Step 2: Subtraction of Vectors

Now, calculate \( \mathbf{u} - \mathbf{v} \).Using \( \mathbf{u} = 2\mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = \mathbf{i} - 3\mathbf{j} \), the difference is:\[ \mathbf{u} - \mathbf{v} = (2\mathbf{i} + \mathbf{j}) - (\mathbf{i} - 3\mathbf{j}) = (2 - 1)\mathbf{i} + (1 + 3)\mathbf{j} = \mathbf{i} + 4\mathbf{j}. \]

Step 3: Dot Product of Vectors

Calculate the dot product \( (\mathbf{u} + \mathbf{v}) \cdot (\mathbf{u} - \mathbf{v}) \).From Step 1 and Step 2, we have \( \mathbf{u} + \mathbf{v} = 3\mathbf{i} - 2\mathbf{j} \) and \( \mathbf{u} - \mathbf{v} = \mathbf{i} + 4\mathbf{j} \). Using the dot product formula \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \), we calculate:\[ (3\mathbf{i} - 2\mathbf{j}) \cdot (\mathbf{i} + 4\mathbf{j}) = (3 \times 1) + (-2 \times 4) = 3 - 8 = -5. \]

Key Concepts

Addition of Vectors

Vector addition involves combining two or more vectors to create a resultant vector. When you add vectors, you simply add their corresponding components. For example, if we have vectors \( \mathbf{a} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{b} = b_1 \mathbf{i} + b_2 \mathbf{j} \), their sum is: \[ \mathbf{a} + \mathbf{b} = (a_1 + b_1) \mathbf{i} + (a_2 + b_2) \mathbf{j}. \] This straightforward operation allows you to visualize vector addition geometrically by placing the tail of one vector at the head of the other.
It is like completing a triangle, where the third side is the vector sum. By applying this to the vectors \( \mathbf{u} \) and \( \mathbf{v} \) from our original exercise, we derived \( 3\mathbf{i} - 2\mathbf{j} \) as the result of their addition.

Subtraction of Vectors

Vector subtraction is similar to vector addition but involves finding the difference between two vectors. The concept is to add one vector to the negative of the other. Thus, for vectors \( \mathbf{a} \) and \( \mathbf{b} \), subtraction is expressed as: \[ \mathbf{a} - \mathbf{b} = (a_1 - b_1) \mathbf{i} + (a_2 - b_2) \mathbf{j}. \] This operation allows you to determine how one vector changes with respect to another, useful in physics for determining displacements.
In the exercise, computing \( \mathbf{u} - \mathbf{v} \) with the given vector components led us to find the result \( \mathbf{i} + 4\mathbf{j} \). This process shows how we effectively reverse the direction of the second vector before adding.

Dot Product

The dot product is a scalar value reflecting the magnitude of projections of one vector onto another. Also known as the scalar product, it is calculated by multiplying corresponding components and adding the products: \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).
Imagine you're projecting the shadow of one vector onto another; that's the dot product.
In the exercise, we multiplied the results from vector addition and subtraction, giving us \((3 \times 1) + (-2 \times 4) = -5\). A negative dot product can indicate the angle between vectors is greater than 90 degrees, demonstrating one vector is pulling in a direction mostly opposite to the other.

Precalculus

Precalculus serves as a bridge between algebra and calculus.
This area of mathematics provides essential tools for understanding functions, limits, and introduces the concept of vectors.
Vectors in precalculus are essential as they set the foundation for vector calculus.
Understanding basics like addition, subtraction, and the dot product of vectors illustrates how quantities can be represented in two or three dimensions. Some key concepts in precalculus include

• Functions and Graphs
• Trigonometry
• Complex Numbers
• Vectors

These concepts are building blocks for calculus and beyond, facilitating a deeper understanding of mathematical and physical problems.