Problem 2 - Advanced Engineering Mathematics

Question

find \(\mathbf{a} \times \mathbf{b}\). $$ a=2 \mathrm{i}+\mathrm{j}, \mathbf{b}=4 \mathbf{i}-\mathbf{k} $$

Step-by-Step Solution

Answer

The cross product is \(-\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}\).

Step 1: Write the Vectors in Component Form

To solve for the cross product \(\mathbf{a} \times \mathbf{b}\), start by writing the vectors \(\mathbf{a}\) and \(\mathbf{b}\) in component form: \(\mathbf{a} = 2\mathbf{i} + 1\mathbf{j} + 0\mathbf{k}\) and \(\mathbf{b} = 4\mathbf{i} + 0\mathbf{j} - 1\mathbf{k}\).

Step 2: Formula for Cross Product

Recall the formula for the cross product of two vectors \(\mathbf{a} \times \mathbf{b}\) given by:\[\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 2 & 1 & 0 \ 4 & 0 & -1 \end{vmatrix}\]This is a determinant that will give us the vector orthogonal to \(\mathbf{a}\) and \(\mathbf{b}\).

Step 3: Compute the Determinant

Calculate the determinant:\[\mathbf{c} = \mathbf{i}(1 \cdot (-1) - 0 \cdot 0) - \mathbf{j}(2 \cdot (-1) - 0 \cdot 4) + \mathbf{k}(2 \cdot 0 - 4 \cdot 1)\]Simplify each component:\[\mathbf{c} = \mathbf{i}(-1) + \mathbf{j}(2) - \mathbf{k}(4)\]Thus, \(\mathbf{c} = -\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}\).

Step 4: Write the Resulting Vector

The vector resulting from the cross product \(\mathbf{a} \times \mathbf{b}\) is \(-\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}\). This vector is perpendicular to both \(\mathbf{a}\) and \(\mathbf{b}\).

Key Concepts

Understanding Vector Operations

Vector operations, such as addition, subtraction, and the cross product, play a crucial role in physics and engineering. While addition and subtraction are relatively straightforward, the cross product can be a bit more complex. The cross product of two vectors, denoted as \( \mathbf{a} \times \mathbf{b} \), results in another vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). This is particularly useful in physics for finding torque or rotational motion when the direction is involved.

• Addition: Combine vectors by adding their corresponding components.
• Subtraction: Subtract vectors by subtracting their corresponding components.
• Cross Product: Use it to find a vector perpendicular to two given vectors.

Vector operations require a good understanding of vector components and how to manipulate them mathematically, which makes them pivotal in many fields where vectors describe quantities.

Calculating the Determinant for Cross Product

The determinant provides a systematic way to find areas, volumes, and in the case of cross products, vector magnitudes in 3D spaces. When computing a cross product, you use a specific determinant formula that captures the perpendicular vector required. This involves setting up a matrix with vectors as rows and solving the determinant. The formula used for two vectors \(\mathbf{a}\) and \(\mathbf{b}\), as shown in the original problem, involves placing the unit vectors \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) in the first row. The components of \(\mathbf{a}\) and \(\mathbf{b}\) fill the next two rows. From here, you calculate the determinant to find the cross product:\[\begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \2 & 1 & 0 \4 & 0 & -1 \end{vmatrix} \]By solving this step-by-step, each sub-determinant provides a component of the resulting vector. This process exemplifies the power of determinants in vector operations, turning matrix calculations into geometric insights.

Properties of Perpendicular Vectors

Perpendicular vectors have a dot product of zero, indicating orthogonality. In geometry, two vectors are perpendicular if they meet at a right angle. This right angle carries significant implications in 3D modeling, physics, and other fields requiring spatial awareness. When you compute the cross product of two vectors, \(\mathbf{a}\) and \(\mathbf{b}\), the resulting vector \(\mathbf{c}\) is perpendicular to both. This property is useful because it provides a standard way to find a direction orthogonal to a plane.

• The dot product of \(\mathbf{c}\) with \(\mathbf{a}\) and \(\mathbf{b}\) should be zero, confirming \(\mathbf{c}\) is perpendicular.
• Applications include calculating normals to surfaces, which are essential in graphics and physics simulations.

Understanding perpendicular vectors enhances your ability to solve problems involving angles, planes, and rotational systems, providing a robust framework for conceptualizing 3D space.