Problem 69 - Precalculus with Calculus Previews

Question

Determine whether the given vectors are orthogonal. $$ \mathbf{u}=\langle-5,-4\rangle, \mathbf{v}=\langle-6,8\rangle $$

Step-by-Step Solution

Answer

The vectors are not orthogonal.

Step 1: Recall the Definition of Orthogonal Vectors

Two vectors are said to be orthogonal if their dot product is zero. The dot product of two vectors \( \mathbf{u} = \langle a, b \rangle \) and \( \mathbf{v} = \langle c, d \rangle \) is given by: \( \mathbf{u} \cdot \mathbf{v} = ac + bd \).

Step 2: Identify the Components of Each Vector

We have two vectors: \( \mathbf{u} = \langle -5, -4 \rangle \) and \( \mathbf{v} = \langle -6, 8 \rangle \). Here, the components of vector \( \mathbf{u} \) are \( a = -5 \) and \( b = -4 \), while the components of vector \( \mathbf{v} \) are \( c = -6 \) and \( d = 8 \).

Step 3: Calculate the Dot Product

Substitute the values of the components into the formula for the dot product: \( \mathbf{u} \cdot \mathbf{v} = (-5)(-6) + (-4)(8) \). Calculate each term: \(-5\times -6 = 30\) and \(-4 \times 8 = -32\). Thus, the dot product is \( 30 + (-32) = -2 \).

Step 4: Determine Orthogonality

Since the dot product of the vectors \( \mathbf{u} \) and \( \mathbf{v} \) is \(-2\), which is not zero, the vectors are not orthogonal.

Key Concepts

Dot Product

The dot product is a fundamental operation in vector algebra that helps us understand the relationship between vectors. It is also known as the scalar product because it results in a single number rather than a vector.
To compute the dot product of two vectors, we multiply corresponding components and then add these products together.

• For two-dimensional vectors, like in our example, the dot product is calculated as: \(\mathbf{u} \cdot \mathbf{v} = ac + bd\)where \(a\) and \(b\) are the components of one vector, and \(c\) and \(d\) are the components of the other vector.
• The result of the dot product can tell us about the angle between the vectors, which is crucial for determining orthogonality.
• If the dot product is zero, the vectors are orthogonal, meaning they are at a right angle to each other.

Understanding the dot product helps us in physics and engineering, where it is used to compute work done by a force and to find projections of vectors.

Vector Components

Every vector in two-dimensional space has two components: one that runs parallel to the x-axis and another that runs parallel to the y-axis.
These components are usually denoted in the form \(\langle a, b \rangle\), where \(a\) is the x-component and \(b\) is the y-component.

• In our exercise, vector \(\mathbf{u}\) is \(\langle -5, -4 \rangle\), giving it x-component \(-5\) and y-component \(-4\).
• Vector \(\mathbf{v}\) is \(\langle -6, 8 \rangle\), giving it x-component \(-6\) and y-component \(8\).

The lengths or magnitudes of these components can give us insights into how the vector moves through space.
Components are a foundation of vector mathematics, allowing us to break down complex motions into simpler, understandable parts.
They show direction and magnitude in each axis, which is key for applications in navigation, physics, and graphics.

Orthogonality

Orthogonality is an essential concept in mathematics, particularly in linear algebra and geometry.
Two vectors are orthogonal if they meet at a right angle, which visually represents the idea of perpendicular lines.
The mathematical condition for orthogonality between two vectors is satisfied when their dot product equals zero.

• This condition implies that the angle between the vectors is exactly 90 degrees.
• Orthogonal vectors are independent of each other in terms of direction, which is useful in many applications, such as signal processing, to ensure two signals do not interfere with each other.

In our exercise, we found that the dot product of vectors \(\mathbf{u}\) and \(\mathbf{v}\) was not zero, leading us to conclude that they are not orthogonal.
Understanding orthogonality helps in simplifying problems in dimensions, designing systems that require distinguishing between different directions or axes, and is fundamental in the construction of coordinate systems.