Problem 44 - Precalculus : Building Concepts and Connections

Question

In this set of exercises, you will use vectors and dot products to study real- world problems. Design The position vectors of a tower and a small garden from the center of a fountain are given by \langle 50,60\rangle and \(\langle 40, y\rangle .\) Find \(y\) so that the two position vectors are orthogonal.

Step-by-Step Solution

Answer

The value of \(y\) that makes the two position vectors orthogonal is -33.33.

Step 1: Understand the concept of orthogonality in vectors

Orthogonality is a fundamental concept in vectors. In a 2-dimensional vector space, two vectors are orthogonal if their dot product equals zero. The dot product is given by multiplying corresponding elements together and summing the results.

Step 2: Formulate the vectors

Given the position vectors of a tower and a small garden from the center of a fountain are \(\langle 50,60\rangle\) and \(\langle 40, y\rangle\). The coordinates x and y of the position vector represent the position in the 2-dimensional plane relative to a reference point, in this case, the center of a fountain.

Step 3: Calculate the dot product

The dot product of two vectors is calculated by multiplying their corresponding components and then adding those products together. The dot product of vectors \(\langle 50,60\rangle\) and \(\langle 40, y\rangle\) is calculated as \(50*40 + 60*y\).

Step 4: Set the dot product to zero and solve for y

To find \(y\) so that the two vectors are orthogonal, the dot product must be equal to zero, Therefore, we set \(50*40 + 60*y = 0\), then simplify and solve for \(y\). This equation turns into \(2000 + 60y = 0\). By subtracting 2000 from both sides and then dividing by 60, we find \(y = -2000 / 60 = -33.33\).

Key Concepts

Understanding the Dot Product

The dot product is a crucial operation in vector mathematics, allowing us to determine relationships between vectors like orthogonality. It's calculated by taking two vectors and multiplying their corresponding components, then adding the results together. For example, with vectors \(\langle a,b \rangle\) and \(\langle c,d \rangle\), the dot product is calculated as \(a \times c + b \times d\). This operation simplifies the process of finding angles between vectors.

• Helps in assessing parallelism and perpendicularity (orthogonality).
• A positive dot product suggests vectors form an acute angle.
• A zero dot product indicates orthogonality.

Understanding the role of the dot product in these mathematical processes is essential in tasks that involve spatial analysis, like in physics and engineering.When calculating dot products, ensure that vector components are aligned properly to avoid errors. Especially when dealing with real-world position vectors as given in the exercise, ensure each component aligns with the correct reference point.

Orthogonality in Vectors: A Key Property

In vector geometry, orthogonality means that two vectors are at right angles to each other. This concept is essential in understanding not just geometry, but also in physics and engineering domains where directions and forces are analyzed.
For vectors in a 2-dimensional space, achieving orthogonality simplifies many calculations in design and analysis. Orthogonal vectors conveniently split spaces into independent dimensions, which is a major advantage:

• They allow for simplicity in expressing vector equations.
• They help in decomposing forces in engineering contexts.
• Orthogonal transformations preserve angles in geometry.

In our exercise, the task was to find a vector component such that two vectors become orthogonal. When two vectors have a dot product of zero—like in our problem where \(\langle 50,60\rangle\) and \(\langle 40,y\rangle\) are solved to be orthogonal—this condition is met, meaning they form a 90° angle. Understanding this condition allows better manipulation and analysis of spatial relationships.

Navigating the 2-Dimensional Vector Space

Vectors in a 2-dimensional space are fundamental in describing geometric and physical phenomena. They have components that position them within this space, having both direction and magnitude.
For instance, the vector \(\langle 50,60 \rangle\) describes a position in the 2-dimensional space relative to a central point. When combined with another vector, say \(\langle 40, y \rangle\), they describe a precise geometric relationship.
Key attributes of 2D vector spaces include:

• Easy representation of real-world scenarios like forces and movement.
• Use as a basis for more complex multidimensional spaces.
• A foundation for complex vector operations like rotations and scaling.

The exercise helps solidify this understanding by asking students to manipulate these vectors till they align orthogonally, illustrating practical use cases of vectors. Developing an intuition for how vectors work within this plane simplifies concepts like the mechanics of movement and force dynamics.