Calculating Scalar Product Of Vectors U = (-4, 17) And V = (2, -11)

by Scholario Team 68 views

In the realm of mathematics, particularly in linear algebra and vector calculus, the scalar product, also known as the dot product, stands as a fundamental operation. It provides a powerful means to understand the relationship between two vectors, offering insights into their magnitudes and the angle between them. This article delves into the intricacies of calculating the scalar product of two-dimensional vectors, using the specific example of vectors u = (-4, 17) and v = (2, -11). We will explore the underlying principles, step-by-step calculations, and the profound implications of the resulting scalar product.

Understanding the Scalar Product

The scalar product is an algebraic operation that takes two vectors as input and returns a single scalar value. This scalar value represents the projection of one vector onto the other, scaled by the magnitude of the second vector. In simpler terms, it quantifies the degree to which two vectors point in the same direction. The scalar product is a fundamental concept in linear algebra, with applications ranging from physics to computer graphics. It provides a way to measure the alignment between vectors, which is crucial in various calculations, such as determining work done by a force or calculating the angle between two lines.

The Formula for Scalar Product Calculation

For two vectors in a two-dimensional space, say u = (u₁, u₂) and v = (v₁, v₂), the scalar product, denoted as u · v, is calculated using the following formula:

u · v = u₁ * v₁ + u₂ * v₂

This formula essentially involves multiplying the corresponding components of the two vectors and summing the results. The simplicity of this calculation belies its profound implications, as the resulting scalar product encapsulates crucial information about the relationship between the vectors.

Step-by-Step Calculation for Vectors u = (-4, 17) and v = (2, -11)

Now, let's apply this formula to the given vectors u = (-4, 17) and v = (2, -11). We will meticulously follow each step to arrive at the scalar product.

  1. Identify the components:

    • u₁ = -4
    • u₂ = 17
    • v₁ = 2
    • v₂ = -11

  2. Apply the formula: u · v = (-4) * (2) + (17) * (-11)

  3. Perform the multiplications: u · v = -8 + (-187)

  4. Sum the results: u · v = -195

Therefore, the scalar product of vectors u and v is -195. This negative value holds significant meaning, which we will explore in the next section.

Interpreting the Scalar Product Result

The resulting scalar product of -195 provides valuable information about the relationship between vectors u and v. The sign and magnitude of the scalar product offer insights into the angle between the vectors and their relative alignment.

The Sign of the Scalar Product

The sign of the scalar product reveals the nature of the angle between the vectors:

  • Positive Scalar Product: Indicates that the angle between the vectors is acute (less than 90 degrees). This means the vectors generally point in the same direction.
  • Zero Scalar Product: Implies that the vectors are orthogonal or perpendicular to each other (angle is 90 degrees). There is no component of one vector in the direction of the other.
  • Negative Scalar Product: Signifies that the angle between the vectors is obtuse (greater than 90 degrees). The vectors generally point in opposite directions.

In our case, the scalar product is -195, which is negative. This tells us that the angle between vectors u and v is obtuse, meaning they point in generally opposite directions.

The Magnitude of the Scalar Product

The magnitude of the scalar product is related to the magnitudes of the individual vectors and the cosine of the angle between them. Specifically:

u · v = |u| |v| cos(Ξ)

where |u| and |v| represent the magnitudes of vectors u and v, respectively, and Ξ is the angle between them. This formula highlights the connection between the scalar product, vector magnitudes, and the angle between the vectors. The magnitude of the scalar product provides a measure of the alignment between the vectors, scaled by their lengths.

In our example, the magnitude of the scalar product is |-195| = 195. This value, combined with the magnitudes of vectors u and v, can be used to calculate the cosine of the angle between them, and subsequently, the angle itself.

Applications of the Scalar Product

The scalar product is not merely a mathematical curiosity; it has profound applications in various fields, including:

Physics

In physics, the scalar product is used to calculate the work done by a force. If a force F acts on an object, causing it to displace by a vector d, the work done W is given by:

W = F · d

This formula elegantly captures the fact that only the component of the force acting in the direction of displacement contributes to the work done. The scalar product also finds applications in calculating power, potential energy, and other physical quantities.

Computer Graphics

In computer graphics, the scalar product is instrumental in determining the lighting and shading of objects. The intensity of light reflected from a surface depends on the angle between the light source and the surface normal (a vector perpendicular to the surface). The scalar product provides a convenient way to calculate the cosine of this angle, which is crucial for shading calculations. Furthermore, the scalar product is used for collision detection and other geometric computations.

Machine Learning

The scalar product plays a crucial role in various machine learning algorithms, particularly in areas like support vector machines (SVMs) and neural networks. In SVMs, the scalar product is used to define the kernel function, which maps data points into a higher-dimensional space where linear separation may be possible. In neural networks, the scalar product is used in the computation of weighted sums of inputs, which form the basis of neuron activation.

Vector Projections

The scalar product is also intimately connected to the concept of vector projections. The projection of vector u onto vector v is a vector that represents the component of u that lies in the direction of v. This projection can be calculated using the scalar product:

projv u = ((u · v) / |v|ÂČ) v

Vector projections are used in various applications, such as resolving forces into components, finding the closest point on a line to a given point, and more.

Conclusion: The Power of the Scalar Product

In conclusion, the scalar product is a fundamental mathematical operation with far-reaching implications. Its ability to quantify the relationship between vectors, particularly their alignment and the angle between them, makes it a powerful tool in various fields. The calculation of the scalar product for vectors u = (-4, 17) and v = (2, -11) resulted in -195, indicating an obtuse angle between the vectors. This example highlights the importance of understanding both the sign and magnitude of the scalar product to fully grasp the relationship between vectors. From physics and computer graphics to machine learning and beyond, the scalar product stands as a testament to the elegance and utility of mathematical concepts in solving real-world problems.

By mastering the scalar product, one gains a deeper understanding of vector algebra and its applications, paving the way for further exploration of advanced mathematical concepts and their diverse applications. Whether you are a student delving into the intricacies of linear algebra or a professional applying mathematical tools in your field, the scalar product is an indispensable concept that will undoubtedly prove invaluable in your endeavors.