Problem 19 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The vectors are perpendicular because their dot product is zero.

Step 1: Understand the definition of perpendicular vectors

Two vectors are perpendicular if their dot product equals zero. We will calculate the dot product of the given vectors \( \mathbf{u} \) and \( \mathbf{v} \) and check if the result is zero.

Step 2: Write the expression for the dot product of two vectors

The dot product of two vectors \( \mathbf{u} = a\mathbf{i} + b\mathbf{j} \) and \( \mathbf{v} = c\mathbf{i} + d\mathbf{j} \) is given by:\[ \mathbf{u} \cdot \mathbf{v} = ac + bd \]

Step 3: Substitute the vector components into the dot product equation

For \( \mathbf{u} = 2\mathbf{i} - 8\mathbf{j} \) and \( \mathbf{v} = -12\mathbf{i} - 3\mathbf{j} \), substitute the components into the dot product equation:\[ \mathbf{u} \cdot \mathbf{v} = (2)(-12) + (-8)(-3) \]

Step 4: Calculate the dot product

Perform the multiplications and additions:\[ \mathbf{u} \cdot \mathbf{v} = (2)(-12) + (-8)(-3) = -24 + 24 \]

Step 5: Determine if the dot product is zero

Since the calculated dot product is:\[ -24 + 24 = 0 \]The dot product is 0, indicating that the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular.

Key Concepts

Understanding the Dot Product

The dot product is a fundamental concept when working with vectors. It is a mathematical operation that takes two equal-length sequences of numbers (usually vectors) and returns a single number. Importantly, the dot product helps us understand the relationship between two vectors—in this case, determining if they are perpendicular.

This scalar product is calculated using the formula:

• For vectors \( \mathbf{u} = a \mathbf{i} + b \mathbf{j} \) and \( \mathbf{v} = c \mathbf{i} + d \mathbf{j} \), their dot product is \( \mathbf{u} \cdot \mathbf{v} = ac + bd \).

When it comes to identifying perpendicular vectors, if the dot product between two vectors is 0, the vectors are perpendicular. This is because a dot product of zero implies that the angle between the vectors is 90 degrees, or a right angle, confirming their perpendicularity.

Diving into Vector Components

Vectors are not just a point with direction; they are broken down into components, often represented in terms of basis vectors \( \mathbf{i} \) and \( \mathbf{j} \). These components help us to calculate things like the dot product.

• For instance, in the vector \( \mathbf{u}=2 \mathbf{i}-8 \mathbf{j} \), "2" is the component along the \( \mathbf{i} \) (or x-axis), and "-8" is along the \( \mathbf{j} \) (or y-axis).
• Similarly, for the vector \( \mathbf{v}=-12 \mathbf{i}-3 \mathbf{j} \), "-12" and "-3" are the components along \( \mathbf{i} \) and \( \mathbf{j} \) respectively.

These components are crucial for performing operations like the dot product. The beauty of vector components lies in their simplicity yet power to convey all necessary information about the vector, from its length to its direction.

Fundamentals of Vector Mathematics

Vector mathematics is a powerful tool in physics and engineering, allowing us to deal with quantities that have both magnitude and direction. It deals with the addition, subtraction, dot products, and several other operations involving vectors.

• Vectors are often represented in a plane using two-dimensional components, but they can be extended into three dimensions or more in advanced contexts.
• The operations carried out with vectors have specific rules—for example, what makes vectors perpendicular is the result of the dot product being zero.

Understanding these principles is key not only in theoretical problems but also in real-world applications, such as forces in equilibrium, projections, and calculations involving torque and momentum. Mastering vector math provides a strong foundation for tackling more complex mathematical and real-world physics problems.