Problem 13 - Algebra and Trigonometry

Question

9–14 Determine whether the given vectors are orthogonal. $$\mathbf{u}=2 \mathbf{i}-8 \mathbf{j}, \quad \mathbf{v}=-12 \mathbf{i}-3 \mathbf{j}$$

Step-by-Step Solution

Answer

The vectors are orthogonal because their dot product is zero.

Step 1: Understand Orthogonal Vectors

Two vectors are orthogonal if their dot product is zero. The dot product of vectors \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \) is calculated as \( a \cdot c + b \cdot d \).

Step 2: Plug in the vector components

For \( \mathbf{u} = 2\mathbf{i} - 8\mathbf{j} \) and \( \mathbf{v} = -12\mathbf{i} - 3\mathbf{j} \), the components are: \( a = 2 \), \( b = -8 \), \( c = -12 \), and \( d = -3 \).

Step 3: Calculate the Dot Product

Substitute the vector components into the dot product formula: \[2 \cdot (-12) + (-8) \cdot (-3) = -24 + 24 = 0.\]

Step 4: Determine Orthogonality

Since the dot product \( 2 \cdot (-12) + (-8) \cdot (-3) = 0 \), the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are orthogonal.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental concept in vector mathematics. It provides a way to multiply two vectors, resulting in a scalar (a single number), rather than another vector. This operation can tell us a lot about the relationship between the two vectors involved.
The formula for the dot product of two vectors \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \) is:

• \( a \cdot c + b \cdot d \)

This means we multiply the corresponding components of each vector and then sum these products.
The dot product can reveal important properties such as orthogonality, which indicates a perpendicular or independent orientation of vectors in their space.

Vector Components

Vector components refer to the parts of a vector that lie along predefined directions, typically along the axes in a coordinate system. In two-dimensional vector spaces, these are often the \( \mathbf{i} \) (x-axis) and \( \mathbf{j} \) (y-axis) components.
To express any vector \( \mathbf{u} \) in components, we use:

• \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \)

Here, \( a \) and \( b \) denote the components of the vector \( \mathbf{u} \) along the \( \mathbf{i} \)-axis and \( \mathbf{j} \)-axis, respectively.
By decomposing vectors into components, we simplify complex vector operations like dot products because we can work separately on each axis. Understanding these components is essential for interpreting how vectors interact through operations such as addition, subtraction, and especially dot products.

Vector Orthogonality

Two vectors are orthogonal if their dot product is zero. Orthogonality indicates that the vectors are perpendicular to each other in a given space. This concept is crucial for numerous mathematical and engineering applications, where perpendicular vectors often represent independent quantities or directions.
For example, given the vectors \( \mathbf{u} = 2\mathbf{i} - 8\mathbf{j} \) and \( \mathbf{v} = -12\mathbf{i} - 3\mathbf{j} \), we check their orthogonality by computing their dot product. If the result is zero, as in

• \( 2 \cdot (-12) + (-8) \cdot (-3) = 0 \)

the vectors are indeed orthogonal.
This property can be especially helpful in designing systems or analyzing data, as it ensures that the effects or dimensions of the vectors are independently oriented from each other.