Problem 15 - Algebra and Trigonometry

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 because their dot product is 0.

Step 1: Recall the definition of perpendicular vectors

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

Step 2: Write down the given vectors

The vectors given in the problem are \( \mathbf{u} = \langle 6, 4 \rangle \) and \( \mathbf{v} = \langle -2, 3 \rangle \). Substitute these values into the formula for the dot product.

Step 3: Calculate the dot product

Compute the dot product \( 6 \cdot (-2) + 4 \cdot 3 \). This simplifies to \( -12 + 12 = 0 \).

Step 4: Conclude whether vectors are perpendicular

Since the dot product \( -12 + 12 = 0 \), the vectors \( \mathbf{u} \) and \( \mathbf{v} \) are perpendicular.

Key Concepts

Understanding the Dot Product

The dot product is a fundamental concept in vector mathematics. It provides a way to measure how parallel or perpendicular two vectors are. In essence, a dot product is a scalar product, meaning it takes two vectors and returns a single number or scalar.
This product is computed by multiplying corresponding components of the vectors and summing the results. For vectors \(\mathbf{u} = \langle a, b \rangle\) and \(\mathbf{v} = \langle c, d \rangle\), the dot product is calculated as:

• \(a \cdot c + b \cdot d\)

One of the significant uses of the dot product is to determine if two vectors are perpendicular. If their dot product equals zero, the vectors are perpendicular.
This condition arises because perpendicular vectors have a 90-degree angle between them, leading to zero cosine in their trigonometric relationship.

Basic Vector Operations

Vector operations form the backbone of understanding vectors. These operations include addition, subtraction, and the famous dot product.
When performing vector addition, the corresponding components of the vectors are summed. Given vectors \(\mathbf{u} = \langle 6, 4 \rangle\) and \(\mathbf{v} = \langle -2, 3 \rangle\), their addition is:

• \(\mathbf{u} + \mathbf{v} = \langle 6 + (-2), 4 + 3 \rangle = \langle 4, 7 \rangle\)

Subtraction follows a similar process, but differences are taken instead.

• \(\mathbf{u} - \mathbf{v} = \langle 6 - (-2), 4 - 3 \rangle = \langle 8, 1 \rangle\)

Operations like these are crucial for manipulating and visualizing vectors. Understanding these properties paves the way for more complex applications, including physics and engineering problems.

Mathematics Education and Vectors

Grasping vectors is a key component of mathematics education. They provide a way to simplify complex problems across various fields like physics, engineering, and computer graphics.
Vectors introduce students to spatial reasoning and thinking beyond numbers to the direction and magnitude. When mathematics education includes vectors, it empowers students with tools that are particularly prevalent in science and technology.
Learning about vectors provides students with the ability to visualize and solve practical problems. For instance, understanding forces acting upon an object, creating computer simulations, and even in graphic modeling. Ultimately, integrating vectors into math education helps build foundational skills in analytical thinking and problem-solving.