Problem 6 - Advanced Engineering Mathematics

Question

find \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=4 \mathbf{i}+\mathbf{j}-5 \mathbf{k}, \mathbf{b}=\mathbf{x}+3 \mathbf{j}-\mathbf{k} $$

Step-by-Step Solution

Answer

\( \mathbf{a} \times \mathbf{b} = 14\mathbf{i} - \mathbf{j} + 11\mathbf{k} \).

Step 1: Identify the Vectors

We are given two vectors: \( \mathbf{a} = 4\mathbf{i} + \mathbf{j} - 5\mathbf{k} \) and \( \mathbf{b} = \mathbf{x} + 3\mathbf{j} - \mathbf{k} \). These vectors are presented in component form where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the unit vectors.

Step 2: Write the Determinant Setup for Cross Product

The cross product \( \mathbf{a} \times \mathbf{b} \) can be calculated using the determinant of a 3x3 matrix where the first row is the unit vectors \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), the second row contains the components of \( \mathbf{a} \), and the third row contains the components of \( \mathbf{b} \): \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 4 & 1 & -5 \ 1 & 3 & -1 \end{vmatrix} \].

Step 3: Calculate the Determinant's Expansion

Expand the determinant by the first row: \[ \mathbf{a} \times \mathbf{b} = \mathbf{i}(1(-1) - (-5)3) - \mathbf{j}(4(-1) - (-5)1) + \mathbf{k}(4(3) - 1(1)). \]

Step 4: Perform the Arithmetic

Calculate each of the components: - The \( \mathbf{i} \) component: \( 1(-1) - (-5)3 = -1 + 15 = 14 \).- The \( \mathbf{j} \) component: \( 4(-1) - (-5)1 = -4 + 5 = 1 \).- The \( \mathbf{k} \) component: \( 4 \times 3 - 1 \times 1 = 12 - 1 = 11 \). Therefore, \( \mathbf{a} \times \mathbf{b} = 14\mathbf{i} - \mathbf{j} + 11\mathbf{k} \).

Step 5: Conclusion

The cross product \( \mathbf{a} \times \mathbf{b} \) is equal to \( 14\mathbf{i} - \mathbf{j} + 11\mathbf{k} \).

Key Concepts

Determinant Expansion

In the context of vector mathematics, obtaining a cross product involves the clever use of determinants. A determinant is a scalar value that is a special number calculated from a square matrix. To find the cross product of two vectors, like in our example with vectors \( \mathbf{a} \) and \( \mathbf{b} \), we represent the cross product using a 3x3 matrix.

This matrix is constructed with its first row as the unit vectors, \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \). The second and third rows are populated by the components of vectors \( \mathbf{a} \) and \( \mathbf{b} \) respectively.

The determinant expansion along the first row—comprising the unit vectors—is necessary to compute the cross product. The method uses minor and cofactor to simplify the matrix, resulting in a vector. Each term in the expanded determinant is associated with a unit vector and computed from a smaller 2x2 determinant. This approach highlights the importance of understanding matrix transformations in vector mathematics.

Vector Mathematics

Vector mathematics involves operations on vectors, which have both magnitude and direction. A vector is often represented in a component form using the unit vectors \( \mathbf{i} \), \( \mathbf{j} \), \( \mathbf{k} \). These represent vectors in the x, y, and z directions in 3D space, respectively.

The cross product of two vectors is particularly distinct because it results in another vector that is perpendicular to the plane formed by the initial vectors. This property is notably different from the dot product, which results in a scalar quantity.

Operations such as addition, subtraction, and scalar multiplication are quite intuitive and involve simple arithmetic on the vector's components. However, the cross product requires a more intricate understanding of angles and geometry, as it combines the aspects of both magnitude and direction uniquely tied to vector mathematics.

Unit Vectors

Unit vectors are fundamental in vector mathematics. They define a direction and are utilized to depict vectors in Cartesian form. Each unit vector has a magnitude of 1 and indicates direction along one of the coordinate axes.

• \( \mathbf{i} \) is the unit vector in the direction of the x-axis.
• \( \mathbf{j} \) is the unit vector in the direction of the y-axis.
• \( \mathbf{k} \) is the unit vector in the direction of the z-axis.

Unit vectors serve an important role in breaking down vectors into their components. When expressing a vector like \( 4\mathbf{i} + \mathbf{j} - 5\mathbf{k} \), this form showcases the vector's influence in each cardinal direction of space.

In the expansion of a determinant for the cross product calculation, unit vectors form the first row of the matrix. This underlines their significance, as they determine the basis for the components of the resulting vector.