Question

Find \(\mathbf{X} \cdot \mathbf{Y}\). (a) \(\mathbf{X}=(3,4,5,-4), \quad \mathbf{Y}=(3,0,3,3)\) (b) \(\mathbf{X}=\left(\frac{1}{6}, \frac{11}{12}, \frac{9}{8}, \frac{5}{2}\right), \quad \mathbf{Y}=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{4},-\frac{1}{4}\right)\) (c) \(\mathbf{X}=(1,2,-3,1,4), \quad \mathbf{Y}=(1,2,-1,3,4)\)

Short answer

Question: Calculate the dot product of the given vectors in the three cases below: (a) \(\mathbf{X}=(3,4,5,-4)\) and \(\mathbf{Y}=(3,0,3,3)\) (b) \(\mathbf{X}=\left(\frac{1}{6}, \frac{11}{12}, \frac{9}{8},\frac{5}{2}\right)\) and \(\mathbf{Y}=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{4},-\frac{1}{4}\right)\) (c) \(\mathbf{X}=(1,2,-3,1,4)\) and \(\mathbf{Y}=(1,2,-1,3,4)\) Answer: (a) The dot product of vectors \(\mathbf{X}\) and \(\mathbf{Y}\) in case (a) is 12. (b) The dot product of vectors \(\mathbf{X}\) and \(\mathbf{Y}\) in case (b) is \(\frac{63}{96}\). (c) The dot product of vectors \(\mathbf{X}\) and \(\mathbf{Y}\) in case (c) is 27.

Step-by-step solution

Case (a)

For the given vectors \(\mathbf{X}=(3,4,5,-4)\) and \(\mathbf{Y}=(3,0,3,3)\), we have: \(\mathbf{X} \cdot \mathbf{Y} = 3 \cdot 3 + 4 \cdot 0 + 5 \cdot 3 + (-4) \cdot 3 = 9 + 0 + 15 - 12 = 12\) So, the dot product of vectors \(\mathbf{X}\) and \(\mathbf{Y}\) in case (a) is 12.

Case (b)

For the given vectors \(\mathbf{X}=\left(\frac{1}{6}, \frac{11}{12}, \frac{9}{8},\frac{5}{2}\right)\) and \(\mathbf{Y}=\left(-\frac{1}{2}, \frac{1}{2}, \frac{1}{4},-\frac{1}{4}\right)\), we have: \(\mathbf{X} \cdot \mathbf{Y} = \frac{1}{6} \cdot \left(-\frac{1}{2}\right) + \frac{11}{12} \cdot \frac{1}{2} + \frac{9}{8} \cdot \frac{1}{4} + \frac{5}{2} \cdot \left(-\frac{1}{4}\right) = -\frac{1}{12}+\frac{11}{24}+\frac{9}{32}-\frac{5}{8}\) Now, finding an equivalent common denominator for all fractions: \(\mathbf{X} \cdot \mathbf{Y} = -\frac{1\cdot8}{12\cdot8}+\frac{11\cdot8}{24\cdot8}+\frac{9\cdot3}{32\cdot3}+\frac{5\cdot6}{8\cdot6} =-\frac{8}{96}+\frac{88}{192}+\frac{27}{96}-\frac{30}{48}\) Now, add the fractions: \(\mathbf{X} \cdot \mathbf{Y} =\frac{-8+88+27\cdot2-30\cdot4}{192} =\frac{-8+88+54-60\cdot2}{192} =\frac{126}{192}\) Finally, simplifying the fraction, we have: \(\mathbf{X} \cdot \mathbf{Y} = \frac{63}{96}\) So, the dot product of vectors \(\mathbf{X}\) and \(\mathbf{Y}\) in case (b) is \(\frac{63}{96}\).

Case (c)

For the given vectors \(\mathbf{X}=(1,2,-3,1,4)\) and \(\mathbf{Y}=(1,2,-1,3,4)\), we have: \(\mathbf{X} \cdot \mathbf{Y} = 1 \cdot 1 + 2 \cdot 2 + (-3) \cdot (-1) + 1 \cdot 3 + 4 \cdot 4 = 1 + 4 + 3 + 3 + 16 = 27\) So, the dot product of vectors \(\mathbf{X}\) and \(\mathbf{Y}\) in case (c) is 27.

Key concepts

vector operations

Vector operations are fundamental tools in mathematics and physics. One of the most common operations is the dot product, also known as the scalar product. This operation involves multiplying corresponding components of two vectors and summing the results. For example, given two vectors \( \mathbf{X} = (x_1, x_2, x_3) \) and \( \mathbf{Y} = (y_1, y_2, y_3) \), the dot product \( \mathbf{X} \cdot \mathbf{Y} \) is calculated as \( x_1y_1 + x_2y_2 + x_3y_3 \).

The dot product result is a scalar, not a vector. It can provide insights such as:

• The angle between the vectors: If the dot product is zero, the vectors are perpendicular.
• The direction of vectors: A positive result indicates similar directions, while a negative result implies opposite directions.
• The magnitude of a vector's projection onto another.

Grasping vector operations, including the dot product, is crucial for solving complex problems in physics, engineering, and computer graphics.

Understanding these operations also forms the basis for more advanced concepts in linear algebra and real analysis.

real analysis

Real analysis is the branch of mathematics that deals with real numbers and real-valued functions. It provides a rigorous foundation for calculus and investigates sequences and series of numbers, their convergence, and the properties of functions and continuity.

In real analysis, vectors and operations like the dot product are essential. The focus on precision and logical structure helps us understand the behavior of vector operations when applied to continuous domains. For example, applying the dot product to real-valued vector functions enables us to analyze the notion of orthogonality and projections within Euclidean spaces.

Some key concepts in real analysis that connect to vector operations include:

• Convergence and limits: Assists in understanding the behavior of sequences of vectors.
• Continuity and differentiability: Helps analyze how small changes in vector inputs affect outputs.
• Metric spaces: Provides a framework for discussing the distance between vectors, which is often used in defining the length and angles in spaces.

Understanding real analysis offers a deeper understanding of spaces and transformations, integral and differential calculus, and more sophisticated applications of vector operations.

linear algebra

Linear algebra is the branch of mathematics that focuses on vector spaces and linear mappings between these spaces. It's a core subject that supports many applied scientific fields. Linear algebra introduces concepts such as vectors, matrices, determinants, and the dot product.

The dot product is a specific operation within linear algebra that serves as a building block for understanding more complex structures such as inner product spaces.

Key areas where linear algebra plays a role include:

• Systems of linear equations: Solving these is one of the central problems in linear algebra. Vectors and their operations, including the dot product, are fundamental in these solutions.
• Matrix transformations: They include operations like rotation, scaling, and translation, all of which can be described using vector operations.
• Eigenvalues and eigenvectors: Eigenvalues are scalars found in linear transformations, and the concept of dot product assists in computations related to them.

Mastery of linear algebra is crucial not only for mathematical theory but also for practical applications. It is essential in computer science, physics, data analysis, and engineering, where vector operations often model real-world systems and transformations.