Problem 13 - Advanced Engineering Mathematics

Question

Find a vector that is perpendicular to both a and b. $$ \mathbf{a}=\mathbf{2} \mathbf{i}+\mathbf{7} \mathbf{j}-\mathbf{4} \mathbf{k}, \mathbf{b}=\mathbf{i}+\mathbf{j}-\mathbf{k} $$

Step-by-Step Solution

Answer

The vector \( \mathbf{a} \times \mathbf{b} = -3\mathbf{i} - 6\mathbf{j} - 5\mathbf{k} \) is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \).

Step 1: Understand the Cross Product Concept

The cross product of two vectors \( \mathbf{a} \) and \( \mathbf{b} \) results in a vector that is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \). To find such a vector, we'll compute the cross product \( \mathbf{a} \times \mathbf{b} \).

Step 2: Set Up the Cross Product Calculation

The vectors \( \mathbf{a} \) and \( \mathbf{b} \) are given by \( \mathbf{a} = 2\mathbf{i} + 7\mathbf{j} - 4\mathbf{k} \) and \( \mathbf{b} = \mathbf{i} + \mathbf{j} - \mathbf{k} \). Write the vectors in component form: \( \mathbf{a} = (2, 7, -4) \) and \( \mathbf{b} = (1, 1, -1) \).

Step 3: Use Determinant Formula for Cross Product

Use the determinant of a 3x3 matrix to find the cross product: \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ 2 & 7 & -4 \ 1 & 1 & -1 \end{vmatrix} \]Calculate the determinant by expanding across the first row.

Step 4: Calculate Determinant for Each Component

Compute the determinant for each component:- \( \, \mathbf{i} ext{-component}: (7)(-1) - (-4)(1) = -7 + 4 = -3 \)- \( \, \mathbf{j} ext{-component}: \text{-}\left((2)(-1) - (-4)(1)\right) = -2 - 4 = -6 \)- \( \, \mathbf{k} ext{-component}: (2)(1) - (7)(1) = 2 - 7 = -5 \)

Step 5: Write the Perpendicular Vector

Combine the computed components to write the vector perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \):\[ \mathbf{a} \times \mathbf{b} = -3\mathbf{i} - 6\mathbf{j} - 5\mathbf{k} \]

Key Concepts

Cross Product

When two vectors are involved, understanding how to compute the cross product is essential. The cross product of two vectors, say \( \mathbf{a} \) and \( \mathbf{b} \), results in a third vector that is perpendicular to both. This is particularly useful in physics and engineering, where a perpendicular vector may represent torque or a normal vector to a plane.
The cross product is denoted by \( \mathbf{a} \times \mathbf{b} \) and is calculated using the components of the vectors. If \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), the resulting cross product vector is determined by a 3x3 determinant as follows:

• \( \mathbf{i} \)-component: \( a_2b_3 - a_3b_2 \)
• \( \mathbf{j} \)-component: \( a_3b_1 - a_1b_3 \)
• \( \mathbf{k} \)-component: \( a_1b_2 - a_2b_1 \)

By calculating each of these components, you find the cross product, which is a vector perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \).

Determinant

The determinant is a critical concept in vector calculus and linear algebra, primarily used in calculating cross products and solving systems of equations. A determinant is a scalar value that is a function of the entries of a square matrix.
For a 3x3 matrix used in cross products, the determinant helps obtain the components of the new vector. The general form for a 3x3 determinant is: \[ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3\end{vmatrix}\] To solve the determinant, expand it across a row (typically the first), using minors and cofactors of the entries. This allows you to derive the components of the resultant vector. Calculating determinants gives insight into the spatial relationships of vectors, such as their relative positions.

Perpendicular Vectors

Perpendicular vectors, also known as orthogonal vectors, have a special property—when two vectors are perpendicular, their dot product is zero. This is because they do not share any "directional" component.
In the context of the cross product, finding a perpendicular vector means finding a vector that makes a right angle (90 degrees) with both of the given vectors. Understanding the geometric implication of perpendicularity is pivotal in fields that rely heavily on spatial representation, such as computer graphics and molecular chemistry.

• Example: If \( \mathbf{a} \) and \( \mathbf{b} \) are vectors in 3D space, the vector \( \mathbf{c} = \mathbf{a} \times \mathbf{b} \) is perpendicular to both \( \mathbf{a} \) and \( \mathbf{b} \).

This concept is vital when designing systems or structures that require perpendicular forces or movements, ensuring stability and efficiency.

3D Geometry

3D geometry involves understanding vectors and their interactions within three-dimensional space. This is a crucial part of vector calculus, where we study not only the direction and magnitude of vectors but also their spatial relationships.
Vectors in 3D geometry can represent various real-world entities like forces, velocities, or accelerations. The cross product, as analyzed, provides a method to find a vector perpendicular to a plane defined by two other vectors, offering great insights into 3D shapes and modeling.

• Visualizing in 3D involves the use of coordinates (x, y, z), and operations like cross products help determine spatial orientation.
• Practical 3D applications include aerospace engineering, where force vectors determine airplane stability, or computer graphics, where normal vectors affect lighting and shading.

In essence, 3D geometry with vector operations enables us to understand and manipulate the spatial aspects of various physical and abstract systems.