Problem 21 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The result is 9.

Step 1: Calculate Dot Product \(\mathbf{u} \cdot \mathbf{v}\)

To find the dot product \( \mathbf{u} \cdot \mathbf{v} \), we multiply the corresponding components of \( \mathbf{u} = 2\mathbf{i} + \mathbf{j} \) and \( \mathbf{v} = \mathbf{i} - 3\mathbf{j} \) and add them together: \( (2)(1) + (1)(-3) = 2 - 3 = -1 \).

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

Next, find the dot product \( \mathbf{u} \cdot \mathbf{w} \). Multiply the corresponding components of \( \mathbf{u} = 2\mathbf{i} + \mathbf{j} \) and \( \mathbf{w} = 3\mathbf{i} + 4\mathbf{j} \): \( (2)(3) + (1)(4) = 6 + 4 = 10 \).

Step 3: Sum the Dot Products

Add the results from the two previous dot products: \( \mathbf{u} \cdot \mathbf{v} + \mathbf{u} \cdot \mathbf{w} = -1 + 10 = 9 \).

Key Concepts

Vector Operations

Vectors are essential in mathematics as they allow us to perform operations that help explain and solve various problems. One key vector operation is the dot product. The dot product is the sum of the products of the corresponding components of two vectors. This operation transforms two vectors into a single scalar value.

For three vectors, such as \( \mathbf{u}, \mathbf{v}, \) and \( \mathbf{w} \), each with two-dimensional components,

• \( \mathbf{u} = 2\mathbf{i} + \mathbf{j} \)
• \( \mathbf{v} = \mathbf{i} - 3\mathbf{j} \)
• \( \mathbf{w} = 3\mathbf{i} + 4\mathbf{j} \)

the dot product helps us find how much one vector aligns with another. This operation is simple yet powerful for calculations involving angles and projections.`

To calculate the dot product, you multiply the corresponding components of two vectors and sum the results. It follows the
formula:\[ \mathbf{u} \cdot \mathbf{v} = u_1 \cdot v_1 + u_2 \cdot v_2 \]
where \( u_1 \) and \( u_2 \) are components of \( \mathbf{u} \), and \( v_1 \) and \( v_2 \) are components of \( \mathbf{v} \).

Algebraic Vectors

Algebraic vectors represent quantities with both magnitude and direction. The algebraic form of a vector, such as \( 2\mathbf{i} + \mathbf{j} \), uses unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) to express directions along the \( x \) and \( y \) axes. In the context of our example, we have:

• \( \mathbf{u} = 2\mathbf{i} + \mathbf{j} \)
• \( \mathbf{v} = \mathbf{i} - 3\mathbf{j} \)
• \( \mathbf{w} = 3\mathbf{i} + 4\mathbf{j} \)

Each vector's components reflect their position and movement direction within the coordinate plane.

Algebraic vectors follow specific rules, making operations like addition and dot product straightforward. Calculating dot products and adding vectors often use the idea of breaking down the vectors by their components.

This approach helps in easily achieving scalar results from operations, allowing users to transition from algebraic to geometric interpretations of vector results. Knowing how to express these calculations in terms of quantities helps in breaking down more complex problems into more manageable parts.

Mathematical Problem Solving

Mathematical problem solving often involves breaking down problems into smaller, more manageable components and understanding how to apply mathematical principles. In our given exercise, we aim to simplify the problem by using the properties of algebraic vectors and vector operations.

The first step involves finding the dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \). We multiply corresponding components to find the result, \(-1\). The second step involves a similar process with vectors \( \mathbf{u} \) and \( \mathbf{w} \), which gives us \(10\).

Finally, adding these results simplifies the problem, leading us to a total of \(9\). This skilled use of vector operations demonstrates how mathematical problem-solving techniques can unravel complex problems into simpler calculations.

Mathematical problem solving allows us to approach concepts like dots products systematically. This understanding is crucial for tackling more advanced areas of physics, engineering, and computer science.