Problem 9 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the dot product of u and v. $$\begin{aligned} &\mathbf{u}=5 \mathbf{i}+\mathbf{j}\\\ &\mathbf{v}=3 \mathbf{i}-\mathbf{j} \end{aligned}$$

Step-by-Step Solution

Answer

The dot product of vectors \( \mathbf{u} \) and \( \mathbf{v} \) is 14.

Step 1: Identify the components of each vector

The vector \( \mathbf{u} = 5\mathbf{i} + \mathbf{j} \) and the vector \( \mathbf{v} = 3\mathbf{i} - \mathbf{j} \). This means that the components of \( \mathbf{u} \) are 5 (for the i component) and 1 (for the j component, as no numerical coefficient is given, we default it to 1), and the components of \( \mathbf{v} \) are 3 (for the i component) and -1 (for the j component, the minus sign indicating a reverse direction).

Step 2: Compute the product of the corresponding components of vectors u and v

For the i-components this will be 5 x 3 = 15, and for the j-components this will be 1 x (-1) = -1.

Step 3: Add the products together

The sum of these products, 15 and -1, gives us the dot product of vectors u and v. So, 15 + (-1) = 14.

Key Concepts

Understanding Vector Components

Vectors are quantities that have both magnitude and direction. To comprehend vectors, it's helpful to break them down into parts, called components. In this problem, the vector \( \mathbf{u} \) is written as \( 5\mathbf{i} + \mathbf{j} \), which means it has a horizontal component of 5 and a vertical component of 1. Similarly, vector \( \mathbf{v} \) is \( 3\mathbf{i} - \mathbf{j} \), with a horizontal component of 3 and a vertical component of -1. Each component is represented in terms of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \), which point in the standard x and y directions respectively.
When dealing with vector problems, accurately breaking a vector into its components is crucial. This allows for easier manipulation and calculation, especially when you are making sense of their geometry or performing operations like dot products.

Mechanics of Vector Multiplication

Vector multiplication can be understood in two primary forms: the dot product and the cross product. Here, we're focusing on the dot product, a way of combining two vectors to produce a scalar value. The dot product method involves multiplying corresponding components together from each vector.

• For the i-components of \( \mathbf{u} \) and \( \mathbf{v} \), multiply 5 and 3 to get 15.
• For the j-components, multiply 1 and -1 to get -1.

This side-by-side multiplication of components makes the concept more straightforward, helping you make calculations that tell us about the orientation and scaling of two vectors in relation to each other.

Summing the Products

Once the components of the vectors have been multiplied, the next step is to find the sum of these numbers. This sum represents the dot product of the two vectors. In this example, we have products of 15 and -1.

• Add 15 and -1 to find the total, which results in 14.

The dot product, in this case, is 14. This value is not just a random number; it provides insight into the relationship between the two vectors, such as angle and direction, or it can even interpret physical quantities like work when dealing with forces.
Summing up individual products is a critical final step in deriving useful information involving vector interactions.