Question

In Exercises 5-10, find the dot product of the given vectors. \(\vec{u}=\langle 2,-4\rangle, \vec{v}=\langle 3,7\rangle\)

Short answer

The dot product is \(-22\).

Step-by-step solution

Understand the Dot Product Formula

The dot product of two vectors \(\vec{u} = \langle a, b \rangle\) and \(\vec{v} = \langle c, d \rangle\) is calculated using the formula: \[\vec{u} \cdot \vec{v} = a \cdot c + b \cdot d\]In this exercise, you need to find the dot product of \(\vec{u} = \langle 2, -4 \rangle\) and \(\vec{v} = \langle 3, 7 \rangle\). This means substituting the values into the formula.

Substitute Values Into the Formula

Substitute the components of the vectors into the dot product formula:- \(a = 2\)- \(b = -4\)- \(c = 3\)- \(d = 7\)So, the dot product is calculated as follows:\[\vec{u} \cdot \vec{v} = 2 \times 3 + (-4) \times 7\]

Perform the Multiplication

Perform the multiplication operations:- Calculate \(2 \times 3 = 6\)- Calculate \((-4) \times 7 = -28\)Thus, we have the expression:\[\vec{u} \cdot \vec{v} = 6 + (-28)\]

Compute the Final Result

Add the results from the multiplication:\[6 + (-28) = 6 - 28 = -22\]So, the dot product of the vectors \(\vec{u}\) and \(\vec{v}\) is \(-22\).

Key concepts

Vectors

Vectors are fundamental elements in mathematics and physics. Think of them as arrows that have both direction and magnitude. They help describe physical quantities that are not just limited to speed, but also include velocity, force, and displacement. Each vector is represented by a set of numerical components, which are essentially coordinates, usually in a two or three-dimensional space. For instance, the vectors given in our problem are represented as \(\vec{u}=\langle 2, -4 \rangle\) and \(\vec{v}=\langle 3, 7 \rangle\). These components specify the position or change in position of the vector in a coordinate system.
Let's go deeper into vector representation:

• The first component signifies the movement along the x-axis.
• The second component highlights movement along the y-axis.

Vectors are an essential part of vector algebra, allowing us to calculate quantities such as the dot product, which is an essential skill in understanding physical phenomena.

Multiplication

In the context of vectors, multiplication often refers to the **dot product**. Understanding this kind of multiplication is crucial when working with vectors. It combines the magnitude and direction of vectors in a specific way. The dot product formula for two vectors \(\vec{u} = \langle a, b \rangle\) and \(\vec{v} = \langle c, d \rangle\) is given by \(\vec{u} \cdot \vec{v} = a \cdot c + b \cdot d\). Notice how it involves multiplying corresponding components, which is different from the typical multiplication of numbers.
Each multiplication has its distinct use in calculating:

• The alignment between two vectors. If the result is zero, the vectors are perpendicular.
• A scalar result, not another vector. This is different from vector cross product.
• Important projections and angles in applications like physics and engineering.

Understanding multiplication in vector terms opens up new ways of analyzing data and resolving spatial problems.

Vector Components

Each vector is defined by its components, which signify its influence in each dimension of the space. The components are base measurements for defining vector length and direction. For example, consider the vector \(\vec{u} = \langle 2, -4 \rangle\):

• Component 2 suggests the influence or movement on the x-axis.
• Component -4 indicates the influence or movement on the y-axis.

The negative sign here shows a direction opposite to the positive axis direction.
Vector components are critical because:

• They allow us to perform calculations and vector operations, like addition and scalar multiplication.
• They help in breaking down more complex vector forms into simpler, usable parts.
• The awareness and understanding of these components form the basis for operations like dot product, where every single component counts towards the final result.

Vector Operations

Vector operations form the backbone of handling vectors in mathematics and physics. They allow the manipulation and transformation of vectors to understand more complex systems. Common operations include addition, subtraction, scalar multiplication, and the dot product.
The **dot product**, in the vector operation context, combines two vectors into a single scalar quantity. This operation is pivotal in determining:

• Whether vectors are parallel, perpendicular, or neither.
• The angle between the vectors, which can inform many practical situations like force alignment.

Performing the dot product involves:

• Multiplying each pair of corresponding components.
• Summing the results to achieve a scalar outcome, as was done to find that \(\vec{u} \cdot \vec{v} = -22\).

By mastering vector operations, students can apply these concepts to academic problems as well as real-world scenarios, simplifying complex notions through fundamental operations.