Angle Between Vectors U = <-2, 3> And V = <-6, 9> Calculation And Explanation

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Determining the angle between vectors is a fundamental concept in linear algebra and vector calculus, with applications spanning various fields such as physics, engineering, and computer graphics. In this article, we will delve into the process of calculating the angle between two given vectors, u = <-2, 3> and v = <-6, 9>. We will explore the underlying principles, the mathematical formula involved, and a step-by-step solution to arrive at the answer. Furthermore, we will discuss the significance of this calculation and its relevance in real-world applications.

Introduction to Vectors and Angles

Before we dive into the specific problem, let's establish a foundational understanding of vectors and angles. A vector is a mathematical object that has both magnitude (length) and direction. It can be represented graphically as an arrow pointing from an initial point to a terminal point. Vectors are often expressed in component form, such as u = <a, b>, where 'a' and 'b' are the components of the vector along the x and y axes, respectively. The magnitude of a vector u = <a, b> is denoted by ||u|| and can be calculated using the Pythagorean theorem: ||u|| = √(a² + b²).

The angle between two vectors is the angle formed when the vectors are placed tail-to-tail. This angle provides valuable information about the relative direction of the vectors. If the angle is 0°, the vectors point in the same direction; if it's 180°, they point in opposite directions; and if it's 90°, they are orthogonal (perpendicular) to each other.

The formula for calculating the angle θ between two vectors u and v is derived from the dot product: u · v = ||u|| ||v|| cos θ. Rearranging this formula, we get:

cos θ = (u · v) / (||u|| ||v||)

θ = arccos[(u · v) / (||u|| ||v||)]

Where:

  • u · v is the dot product of vectors u and v.
  • ||u|| is the magnitude of vector u.
  • ||v|| is the magnitude of vector v.
  • arccos is the inverse cosine function.

This formula is crucial for determining the angle between any two vectors, and we will apply it to the given vectors in the following sections.

Step-by-Step Solution: Finding the Angle Between u = <-2, 3> and v = <-6, 9>

Now, let's apply the formula to find the angle between the vectors u = <-2, 3> and v = <-6, 9>. We will follow a step-by-step approach to ensure clarity and accuracy.

Step 1: Calculate the Dot Product (u · v)

The dot product of two vectors u = <u₁, u₂> and v = <v₁, v₂> is calculated as follows:

u · v = u₁v₁ + u₂v₂

For our vectors u = <-2, 3> and v = <-6, 9>:

u · v = (-2)(-6) + (3)(9) = 12 + 27 = 39

Therefore, the dot product of u and v is 39. This scalar value represents the projection of one vector onto the other, scaled by the magnitude of the second vector. A positive dot product indicates that the angle between the vectors is acute (less than 90°), while a negative dot product indicates an obtuse angle (greater than 90°). A dot product of zero means the vectors are orthogonal.

Step 2: Calculate the Magnitudes of u and v

The magnitude of a vector u = <a, b> is calculated as:

||u|| = √(a² + b²)

For vector u = <-2, 3>:

||u|| = √((-2)² + (3)²) = √(4 + 9) = √13

For vector v = <-6, 9>:

||v|| = √((-6)² + (9)²) = √(36 + 81) = √117 = √(9 * 13) = 3√13

The magnitudes of u and v represent their lengths. The magnitude of u is √13, and the magnitude of v is 3√13. These values are essential for normalizing the vectors and calculating the cosine of the angle between them.

Step 3: Calculate the Cosine of the Angle (cos θ)

Using the formula cos θ = (u · v) / (||u|| ||v||), we can now calculate the cosine of the angle between u and v:

cos θ = 39 / (√13 * 3√13) = 39 / (3 * 13) = 39 / 39 = 1

The cosine of the angle between u and v is 1. This value is crucial because it directly relates to the angle itself. A cosine of 1 indicates that the angle between the vectors is 0°, which means the vectors point in the same direction.

Step 4: Find the Angle θ

To find the angle θ, we take the inverse cosine (arccos) of the value we calculated in the previous step:

θ = arccos(1)

The arccos(1) is 0°.

Therefore, the angle between the vectors u = <-2, 3> and v = <-6, 9> is 0°. This result confirms that the vectors are pointing in the same direction, which is also evident from the fact that v is a scalar multiple of u (v = 3u).

Deeper Dive: Understanding the Implications of the Angle Between Vectors

The angle between vectors is not just a mathematical curiosity; it has significant implications in various fields. Understanding this concept allows us to analyze the relationships between different quantities represented as vectors.

Parallel and Anti-Parallel Vectors

When the angle between two vectors is 0°, the vectors are parallel and point in the same direction. This means that one vector is a scalar multiple of the other, as we observed in our example with u = <-2, 3> and v = <-6, 9>. Parallel vectors are essential in physics for describing forces acting in the same direction or velocities of objects moving along the same path.

Conversely, when the angle between two vectors is 180°, the vectors are anti-parallel and point in opposite directions. This situation often arises when dealing with opposing forces or velocities. For instance, the force of friction opposes the direction of motion, resulting in an angle of 180° between the force and velocity vectors.

Orthogonal Vectors

If the angle between two vectors is 90°, the vectors are orthogonal, or perpendicular, to each other. Orthogonality is a crucial concept in linear algebra and has applications in various areas, including computer graphics, signal processing, and data analysis. In computer graphics, orthogonal vectors are used to define coordinate systems and perform transformations. In signal processing, orthogonal functions are used to decompose signals into their constituent frequencies.

Applications in Physics and Engineering

In physics, the angle between vectors is used to calculate work done by a force. Work is defined as the dot product of the force and displacement vectors, which involves the cosine of the angle between them. If the force and displacement are in the same direction (angle 0°), the work done is maximum. If they are perpendicular (angle 90°), no work is done.

In engineering, the angle between vectors is used in structural analysis to determine the forces acting on a structure. Engineers use vector analysis to ensure that structures are stable and can withstand the applied loads. The concept of orthogonality is also used in designing efficient electrical circuits and communication systems.

Relevance in Computer Graphics and Game Development

In computer graphics and game development, vectors are used to represent positions, directions, and orientations. The angle between vectors is used to calculate lighting effects, collision detection, and object interactions. For example, the angle between a light source vector and a surface normal vector determines the intensity of light reflected from the surface. In collision detection, the angle between the velocity vectors of two objects can be used to determine the outcome of a collision.

Conclusion: The Significance of Understanding Angles Between Vectors

In conclusion, determining the angle between vectors is a fundamental concept with far-reaching applications. By understanding the formula and the underlying principles, we can analyze the relationships between vectors and apply this knowledge to solve problems in various fields. In the specific case of vectors u = <-2, 3> and v = <-6, 9>, we found that the angle between them is 0°, indicating that they are parallel and point in the same direction. This understanding is crucial for anyone working with vectors, whether in mathematics, physics, engineering, or computer science. The ability to calculate and interpret the angle between vectors empowers us to model and analyze complex systems, making it an invaluable skill in the modern world.