Problem 49 - Precalculus with Calculus Previews

Question

In Problems \(49-62, \mathbf{u}=\langle 2,-3\rangle, \mathbf{v}=\langle-1,5\rangle,\) and \(\mathbf{w}=\langle 3,-2\rangle .\) Find the indicated scalar or vector. $$ {u} \cdot \mathbf{v} $$

Step-by-Step Solution

Answer

The dot product \( \mathbf{u} \cdot \mathbf{v} = -17 \).

Step 1: Recall the Dot Product Formula

The dot product of two vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \) is given by \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \).

Step 2: Apply the Formula to Given Vectors

For the vectors \( \mathbf{u} = \langle 2, -3 \rangle \) and \( \mathbf{v} = \langle -1, 5 \rangle \), substitute their components into the dot product formula: \( \mathbf{u} \cdot \mathbf{v} = (2)(-1) + (-3)(5) \).

Step 3: Calculate the Products

Compute the products: \( (2)(-1) = -2 \) and \( (-3)(5) = -15 \).

Step 4: Sum the Results

Add the results from the previous step: \( -2 + (-15) = -17 \).

Key Concepts

Vectors

In mathematics, vectors are fundamental entities used to represent quantities that have both magnitude and direction. Think of a vector as an arrow pointing in space. The length of the arrow signifies the magnitude while the direction it points to describes its direction. Vectors are essential in various subjects including physics, engineering, and computer graphics as they can efficiently represent forces, velocities, and many other physical quantities.

When working with vectors, we often express them in terms of their components. In a two-dimensional space, a vector can be represented as \( \mathbf{u} = \langle x, y \rangle \), where \( x \) and \( y \) are the components of the vector. These components dictate the vector's position relative to an origin in the coordinate plane.

Moreover, vectors can be added together, scaled, or involved in more complex operations such as calculating the dot product, which creates a more profound understanding of their interactions.

Vector Components

Understanding vector components is crucial for performing operations with vectors. A vector component breaks down a vector into its individual horizontal and vertical parts or directions. Using these components, you can reconstruct the vector entirely. For example, if you have a vector like \( \mathbf{u} = \langle 2, -3 \rangle \), the number 2 represents how far along the x-axis the vector moves, while -3 indicates the movement along the y-axis.

The components can be used in various operations such as determining the angle of the vector, its length (or magnitude), and in calculating other mathematical products like the dot product. Knowing this, you can easily manipulate vectors to fit the requirements of specific mathematical or physical scenarios.

The beauty of vector components lies in their simplicity. Despite dealing with complex physical phenomena, breaking vectors down into components provides clarity and ease when it comes to calculations and visualizations.

Mathematical Operations

One key mathematical operation with vectors is the dot product. The dot product offers insight into the relationship between two vectors without much spatial fuss. It produces a scalar – a single number – that tells us how two vectors "align" with each other. This operation is vital in physics for calculating work done by a force and in algorithms to find the angle between two vectors.

To compute the dot product, if you have vectors \( \mathbf{a} = \langle a_1, a_2 \rangle \) and \( \mathbf{b} = \langle b_1, b_2 \rangle \), you apply the formula: \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \). This calculation involves multiplying the corresponding components from each vector and then adding the results. For our example vectors \( \mathbf{u} = \langle 2, -3 \rangle \) and \( \mathbf{v} = \langle -1, 5 \rangle \), it becomes \( 2 \times -1 + (-3) \times 5 = -2 - 15 = -17 \).

This resultant scalar is incredibly useful not just for theoretical purposes but also for solving real-world problems, as it quantifies how 'aligned' the vectors are.