Problem 15 - Precalculus Mathematics for Calculus

Question

Determine whether the given vectors are perpendicular. $$\mathbf{u}=\langle 6,4\rangle, \quad \mathbf{v}=\langle- 2,3\rangle$$

Step-by-Step Solution

Answer

The vectors are perpendicular as their dot product is zero.

Step 1: Understand Perpendicularity of Vectors

Two vectors are perpendicular if their dot product equals zero.

Step 2: Find the Dot Product of the Vectors

The dot product of two vectors \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \) is calculated as \( u_1 \cdot v_1 + u_2 \cdot v_2 \). Plug in the values of \( \mathbf{u} = \langle 6,4 \rangle \) and \( \mathbf{v} = \langle -2,3 \rangle \).

Step 3: Calculate the Dot Product

Substitute the components into the dot product formula: \( 6 \cdot (-2) + 4 \cdot 3 = -12 + 12 \).

Step 4: Evaluate the Dot Product

Calculate \( -12 + 12 \) which equals \( 0 \). This confirms the vectors are perpendicular.

Key Concepts

Dot Product

Understanding the dot product is crucial when dealing with vector mathematics. In simple terms, the dot product is a way to multiply two vectors and get a scalar (a single number) as a result.
To compute the dot product of two vectors, \( \mathbf{u} = \langle u_1, u_2 \rangle \) and \( \mathbf{v} = \langle v_1, v_2 \rangle \), you use the formula:

• \( u_1 \cdot v_1 + u_2 \cdot v_2 \)

This means you multiply the corresponding components of the vectors and add the results.
The dot product plays a key role in determining the angle between vectors and is zero if the vectors are perpendicular. This is because a zero dot product implies the cosine of the angle between the vectors is zero, which corresponds to a right angle. Understanding this concept helps in many practical applications, like projecting vectors onto one another or finding angles between directions.

Perpendicular Vectors

Perpendicular vectors form a right angle with each other. In vector mathematics, two vectors are considered perpendicular if their dot product is zero. This is because the dot product is closely related to the cosine of the angle between the vectors.
When the dot product equals zero, the angle between the vectors is 90 degrees. Here’s a quick way to remember:

• A zero dot product means perpendicular vectors.
• If the dot product is positive, the vectors are more aligned.
• If negative, the vectors point in opposite directions.

Using these concepts allows you to quickly determine relationships between vectors in physics and engineering, such as ensuring structural components or forces are properly aligned or oppose each other effectively.

Vector Components

Vector components are the essential building blocks of vectors, representing the vector's influence in each dimension. A vector \( \mathbf{u} = \langle u_1, u_2 \rangle \) is expressed through its components \( u_1 \) and \( u_2 \), typically signifying horizontal and vertical influences if you’re in a two-dimensional space.
Calculating the effect or interaction of vectors relies heavily on these components. This involves:

• Breaking vectors into their components according to axes.
• Using these components in mathematical operations like addition, subtraction, and the dot product.

By understanding components, you can dissect and analyze how vectors behave, allowing you to solve complex problems involving movement, forces, and directions in multiple dimensions.