Problem 1 - Advanced Engineering Mathematics

Question

find \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=\mathbf{i}-\mathbf{j}, \mathbf{b}=\mathbf{3} \mathbf{j}+5 \mathbf{k} $$

Step-by-Step Solution

Answer

The cross product is \(-5\mathbf{i} - 5\mathbf{j} + 3\mathbf{k}\).

Step 1: Recall the Cross Product Formula

Recall that the cross product of two vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k} \) is given by:\[\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}\]This formula requires users to calculate the determinant of the matrix.

Step 2: Identify the Components of the Vectors

For the given vectors, identify each of the components:- \( \mathbf{a} = \mathbf{i} - \mathbf{j} \) translates to \( a_1 = 1, a_2 = -1, a_3 = 0 \).- \( \mathbf{b} = 3\mathbf{j} + 5\mathbf{k} \) translates to \( b_1 = 0, b_2 = 3, b_3 = 5 \).

Step 3: Set Up the Determinant

Using the components from Step 2, set up the determinant to calculate the cross product:\[\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 1 & -1 & 0 \ 0 & 3 & 5 \end{vmatrix}\]

Step 4: Calculate the Determinant

Expand the determinant along the first row:\[\mathbf{a} \times \mathbf{b} = \mathbf{i}( (-1)(5) - (0)(3) ) - \mathbf{j}( (1)(5) - (0)(0) ) + \mathbf{k}( 1(3) - (0)(-1) )\]This simplifies to:\[\mathbf{i}(-5) - \mathbf{j}(5) + \mathbf{k}(3)\]

Step 5: Simplify the Expression

Simplify the expression obtained in the previous step:\[\mathbf{a} \times \mathbf{b} = -5\mathbf{i} - 5\mathbf{j} + 3\mathbf{k}\]This is the cross product of \( \mathbf{a} \) and \( \mathbf{b} \).

Key Concepts

Vector Calculus

Vector calculus is a branch of mathematics that focuses on differentiation and integration of vector fields. It's a critical component in physics and engineering, as it helps analyze physical quantities that have both magnitude and direction, such as force and velocity. One of the fundamental operations in vector calculus is the cross product, which helps determine a vector perpendicular to two given vectors. This perpendicularity is key in understanding concepts like torque and rotation in physics, where the direction of effects is as crucial as their magnitudes. When dealing with vectors, especially in vector calculus, it’s important to break them down into components to perform operations like cross products. This allows for the systematic manipulation of vectors using algebraic rules, leading to accurate calculations in applied sciences. Understanding these principles is essential for students pursuing fields that heavily rely on mathematics and physics.

Determinants

Determinants play a crucial role in linear algebra and vector calculus, especially when calculating cross products. A determinant is a scalar value that can be computed from the elements of a square matrix. It's useful for solving systems of linear equations, finding eigenvalues, and even in calculus operations involving matrix transformations. In the context of the cross product, the determinant format helps establish a systematic way of computing a vector that is orthogonal to two given vectors. By arranging vectors in a 3x3 matrix with unit vectors in the first row, the determinant calculation ensures an accurate result. This process involves cofactor expansion along a row or column, commonly done along the top row in a cross product computation:

• Multiply along diagonals to find minor determinants.
• Use cofactor signs to adjust the computed minors.
• Add up the adjusted minors to find the cross product in vector form.

Understanding determinants provides more insight into how transformations affect vector spaces, and it equips students with vital analytic tools for more complex vector operations.

Vector Operations

In vector calculus, operations like addition, subtraction, and multiplication of vectors (including dot and cross products) are fundamental. These operations help describe and solve problems involving forces in physics, movements in robotics, or any scenario where geometric representation is useful. The cross product of two vectors yields another vector that is perpendicular to both original vectors. The resulting vector's direction is determined by the right-hand rule, and its magnitude is given by the product of the magnitudes of the original vectors and the sine of the angle between them.

• Cross products are used to find normal vectors to surfaces.
• They help calculate torque in rotational motion.
• They determine the area of parallelograms formed by two vectors.

Proficiency in vector operations enables students to efficiently navigate disciplines requiring spatial reasoning and vector manipulation, making it a vital concept in mathematical education.

Mathematics Education

Mathematics education aims to provide students with the tools and knowledge they need to understand and apply mathematical concepts across various areas. Topics like vector calculus are crucial for students in STEM (Science, Technology, Engineering, and Mathematics) fields, highlighting the need for robust educational techniques. Teaching vector operations, especially cross products, not only involves explaining the algorithms and methods used but also ensuring students grasp the physical intuition behind the math. Techniques include:

• Using visual aids to explain vector directions and magnitudes.
• Providing real-world examples where cross products apply, like in physics or engineering.
• Incorporating technology, such as graphing calculators and computer software, to visualize concepts.

By focusing on practical applications and conceptual understanding, mathematics educators can enhance student learning and equip them with the skills necessary for innovative careers in STEM.