Problem 17 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The vectors are not perpendicular because their dot product is 4.

Step 1: Understand Perpendicular Vectors

Vectors are perpendicular 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 calculated as \(a \cdot c + b \cdot d\).

Step 2: Calculate the Dot Product

Compute the dot product of \(\mathbf{u} = \langle -2, 6 \rangle\) and \(\mathbf{v} = \langle 4, 2 \rangle\). Substitute the components into the dot product formula: \[(-2) \cdot 4 + 6 \cdot 2\]Calculate each part: \(-2 \cdot 4 = -8\) and \(6 \cdot 2 = 12\). Add the results: \(-8 + 12 = 4\).

Step 3: Analyze the Dot Product

The dot product of the vectors \(\mathbf{u}\) and \(\mathbf{v}\) is 4, which is not zero. Therefore, the vectors are not perpendicular.

Key Concepts

Dot Product

The dot product is a fundamental concept in vector mathematics. It allows us to calculate a scalar that reveals important information about the relationship between vectors. To find the dot product of two vectors, say \(\mathbf{u} = \langle a, b \rangle\) and \(\mathbf{v} = \langle c, d \rangle\), you use the formula: \[a \cdot c + b \cdot d\]. This results in a single number.

By calculating the dot product, we can determine if vectors are perpendicular. If the dot product is zero, the vectors are perpendicular. In simpler terms, perpendicular vectors meet at a 90-degree angle. Knowing how to calculate the dot product helps in various mathematical fields, including physics and engineering, where vector interactions are crucial.

Vectors

Vectors are mathematical entities possessing both magnitude and direction. Imagine them as arrows that not only stretch in space but also point in a specific direction. This makes them highly versatile, especially in fields that require the representation of multidimensional data.

• **Magnitude**: Represents how long the vector is (think of the length of an arrow).
• **Direction**: Describes the way in which the arrow points.

When dealing with vectors like \(\mathbf{u} = \langle -2, 6 \rangle\) and \(\mathbf{v} = \langle 4, 2 \rangle\), it’s their interplay through operations such as the dot product that reveals specific properties like perpendicularity. Understanding vectors is essential for geometric interpretations and solving real-world problems that involve forces or velocities.

Mathematics

Mathematics is a broad field that encompasses several concepts and operations, including vectors and their dot products. In mathematical terms, operations with vectors help solve complex problems by simplifying and providing precise quantitative insights. It’s this capability that makes mathematics a universal language in science and technology.

When studying vectors in mathematics, we often assess their relationships and interactions using operations like the dot product. This operation allows us to:

• Determine perpendicularity.
• Calculate projection lengths.
• Understand force directions in physics.

Thus, mathematics gives us the tools to understand and model the real world more effectively. By learning the foundational elements like dot products and vectors, students are better equipped to tackle complex mathematical problems with confidence.