Problem 16 - Algebra and Trigonometry

Question

Let $$\mathbf{u}=2 \mathbf{i}-\mathbf{j}, \quad \mathbf{v}=3 \mathbf{i}+\mathbf{j}, \quad \text { and } \quad \mathbf{w}=\mathbf{i}+4 \mathbf{j}$$ Find each specified scalar. $$ 5(\mathbf{v} \cdot \mathbf{w}) $$

Step-by-Step Solution

Answer

The specified scalar is 35

Step 1: Compute the dot product \(\mathbf{v} \cdot \mathbf{w}\)

To compute the dot product of two vectors, multiply the corresponding components of the vectors and sum them. Given \(\mathbf{v} = 3\mathbf{i} + \mathbf{j}\) and \(\mathbf{w} = \mathbf{i} + 4\mathbf{j}\), we have \( \mathbf{v} \cdot \mathbf{w} = (3 \times 1) + (1 \times 4) = 3 + 4 = 7\)

Step 2: Multiply by Scalar

Multiply the result of the dot product by the given scalar. In this case, the scalar is 5. Performing this calculation gives us the final result: \(5 \times (\mathbf{v} \cdot \mathbf{w}) = 5 \times 7 = 35\)

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a way of multiplying two vectors, resulting in a scalar value. It involves:

• Multiplying the corresponding components of two vectors.
• Summing up these individual products to get a single number.

This operation is only defined for two vectors of the same dimensions. Let's take the example of vectors \( \mathbf{v} \) and \( \mathbf{w} \) from our exercise:

Given that \( \mathbf{v} = 3\mathbf{i} + \mathbf{j} \) and \( \mathbf{w} = \mathbf{i} + 4\mathbf{j} \), the dot product is calculated as follows:
\[ \mathbf{v} \cdot \mathbf{w} = (3 \times 1) + (1 \times 4) = 3 + 4 = 7 \]
This result, 7, is the scalar outcome of the dot product operation.

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a scalar (a single number), which then scales the vector. Here’s the basic idea:

• Each component of the vector is multiplied by the scalar.
• The direction of the vector remains unchanged unless the scalar is negative, in which case it reverses direction.

In the original exercise, after computing the dot product, we are given a scalar to multiply this result. With the scalar being \( 5 \) and the dot product \( 7 \), the calculation is straightforward:
\[ 5 \times 7 = 35 \]
This results in a new scalar value of 35, which is the final answer to the problem.

Vectors

A vector is a mathematical entity that has both a magnitude and a direction. Vectors are essential in physics and engineering to represent quantities such as velocity and force. In two-dimensional space, vectors can be represented as a linear combination of unit vectors \( \mathbf{i} \) and \( \mathbf{j} \):
\[ \mathbf{v} = x\mathbf{i} + y\mathbf{j} \]
Here, \( x \) and \( y \) are the vector components. The set of vectors in our exercise \( \mathbf{u} = 2 \mathbf{i} - \mathbf{j} \), \( \mathbf{v} = 3\mathbf{i} + \mathbf{j} \), and \( \mathbf{w} = \mathbf{i} + 4\mathbf{j} \) illustrate how they can be combined to compute operations such as the dot product.

Vectors help in simplifying complex geometrical and physical problems by allowing operations like addition, subtraction, and multiplication by scalars to be performed easily.

Vector Components

The components of a vector are the projections of that vector along the coordinate axes. In a 2D space, any vector \( \mathbf{v} \) can be broken down into:

• An \( x \)-component along the horizontal axis.
• A \( y \)-component along the vertical axis.

For example, the vector \( \mathbf{v} = 3\mathbf{i} + \mathbf{j} \) has a horizontal component of 3 and a vertical component of 1. Components make it easier to perform mathematical operations like the dot product, since you can handle each dimension separately.

Understanding the components of a vector is crucial to mastering vector arithmetic, as they form the building blocks for more advanced calculations and concepts, such as determining a vector's magnitude and direction.