Problem 10 - Advanced Engineering Mathematics

Question

find \(\mathbf{a} \times \mathbf{b}\). $$ \mathbf{a}=(8,1,-6), \mathbf{b}=\langle 1,-2,10\rangle $$

Step-by-Step Solution

Answer

The cross product is \((-2, -86, -17)\).

Step 1: Identify Components

Vectors \( \mathbf{a} \) and \( \mathbf{b} \) are given in component form. \( \mathbf{a} = (8, 1, -6) \) means the components are \( a_1 = 8 \), \( a_2 = 1 \), \( a_3 = -6 \). Similarly, \( \mathbf{b} = (1, -2, 10) \) gives \( b_1 = 1 \), \( b_2 = -2 \), and \( b_3 = 10 \).

Step 2: Use Cross Product Formula

The cross product \( \mathbf{a} \times \mathbf{b} \) is given by the determinant: \[ \mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \] Where \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the unit vectors in the x, y, and z directions.

Step 3: Calculate the Determinant

Expand the determinant using the method of minors and cofactors:\[ \mathbf{a} \times \mathbf{b} = \mathbf{i}(a_2b_3 - a_3b_2) - \mathbf{j}(a_1b_3 - a_3b_1) + \mathbf{k}(a_1b_2 - a_2b_1) \] Substituting the component values:\[ = \mathbf{i}(1 \times 10 - (-6) \times (-2)) - \mathbf{j}(8 \times 10 - (-6) \times 1) + \mathbf{k}(8 \times (-2) - 1 \times 1) \]

Step 4: Simplify Each Term

Calculate the simplified forms of each component:\[ \mathbf{i}(10 - 12) = \mathbf{i}(-2) \]\[ - \mathbf{j}(80 + 6) = -\mathbf{j}(86) \]\[ \mathbf{k}(-16 - 1) = \mathbf{k}(-17) \]

Step 5: Combine Results

Combine the individual components into a single vector:\[ \mathbf{a} \times \mathbf{b} = (-2)\mathbf{i} - 86\mathbf{j} - 17\mathbf{k} \]This gives us \( \mathbf{a} \times \mathbf{b} = (-2, -86, -17) \).

Key Concepts

Vector Mathematics

Vector mathematics is an important topic in physics and engineering, where quantities are represented by vectors. Vectors have both magnitude and direction, making them different from scalar quantities, which only have magnitude.

In the context of our problem, we are dealing with two 3-dimensional vectors, \( \mathbf{a} = (8, 1, -6) \) and \( \mathbf{b} = (1, -2, 10) \). The cross product is a binary operation on two vectors in three-dimensional space that results in another vector.

Some key features of vector cross product include:

• It results in a vector that is perpendicular to both original vectors.
• Its magnitude is given by the area of the parallelogram that the vectors span.
• The direction of the resultant vector is determined by the right-hand rule.

Understanding how to perform vector operations like the cross product can aid in solving complex physical problems, such as determining torques and rotational forces.

Determinant Calculation

To calculate the cross product of the vectors \( \mathbf{a} \) and \( \mathbf{b} \), we use a determinant involving unit vectors. The determinant allows us to compute the cross product in a systematic manner.

The determinant is structured as follows:\[ \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix} \]Here, \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the standard unit vectors in the x, y, and z directions, respectively.

The calculation involves expanding the determinant using a method called expansion by minors and cofactors. Each term of the cross product corresponds to a minor determinant:

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

When solved, this determinant gives us the new vector \( \mathbf{a} \times \mathbf{b} = (-2, -86, -17) \), vividly illustrating the power and utility of using determinants in vector mathematics.

Unit Vectors

In three-dimensional space, unit vectors are essential as they provide a basis for describing any vector. Each unit vector has a magnitude of 1 and points along the axes of a coordinate system.

There are three primary unit vectors in a Cartesian coordinate system:

• \( \mathbf{i} \): points in the positive x-direction
• \( \mathbf{j} \): points in the positive y-direction
• \( \mathbf{k} \): points in the positive z-direction

In the cross product computation, unit vectors are positioned in the determinant to symbolize the direction of each component of the resulting vector.

They ensure that each component of a vector is accounted for along the x, y, and z axes. The use of unit vectors provides standardization, making vector calculations uniform. Understanding unit vectors is crucial since they act as building blocks for more complex vector expressions and operations.