Problem 31 - Advanced Engineering Mathematics

Question

Find the indicated scalar or vector without using \((5),(13)\), or \((15)\). $$ \mathbf{i} \times(\mathbf{i} \times \mathbf{j}) $$

Step-by-Step Solution

Answer

The result is \(-\mathbf{j}\).

Step 1: Recognize Vector Triple Product Identity

The expression \( \mathbf{i} \times (\mathbf{i} \times \mathbf{j}) \) involves a vector triple product. We need to recall the vector identity: \[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \]. This identity will help us simplify the expression without directly using equations (5), (13), or (15).

Step 2: Identify the Vectors

In the problem \( \mathbf{a} = \mathbf{i} \), \( \mathbf{b} = \mathbf{i} \), and \( \mathbf{c} = \mathbf{j} \).

Step 3: Calculate Dot Products

Compute the dot products:\[ \mathbf{a} \cdot \mathbf{c} = \mathbf{i} \cdot \mathbf{j} = 0 \]\[ \mathbf{a} \cdot \mathbf{b} = \mathbf{i} \cdot \mathbf{i} = 1 \]

Step 4: Substitute in Vector Triple Product Identity

Substitute the dot products from Step 3 into the vector identity:\[ \mathbf{i} \times (\mathbf{i} \times \mathbf{j}) = (0)\mathbf{i} - (1)\mathbf{j} = -\mathbf{j} \]

Step 5: Final Result

The expression \( \mathbf{i} \times(\mathbf{i} \times \mathbf{j}) \) simplifies to \(-\mathbf{j}\).

Key Concepts

Vector Triple Product

The Vector Triple Product is a concept in vector algebra involving the cross product of one vector with the cross product of two other vectors. It generally follows the format of \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) \). Understanding this product is crucial in various fields of physics and engineering, where vectors are used to represent forces, velocities, and more.

There is a specific vector identity that simplifies the calculation of the vector triple product:

• \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \)

This formula shows that the vector triple product of \( \mathbf{a}, \mathbf{b}, \text{ and } \mathbf{c} \) is equivalent to combining two dot products and the original vectors in a specific way, which helps avoid complicated multiplication. By applying this identity, the computation becomes more intuitive and less error-prone.

Cross Product

The Cross Product of two vectors produces another vector which is perpendicular to both original vectors. This specific operation is unique to three-dimensional space and follows certain rules that ensure the resulting vector has a direction according to the right-hand rule and a magnitude based on the area of the parallelogram formed by the original vectors.

The cross product formula is expressed as:

• \( \mathbf{a} \times \mathbf{b} = \left|\begin{array}{ccc}\mathbf{i} & \mathbf{j} & \mathbf{k}\ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{array}\right| \)

The cross product is anticommutative, meaning \( \mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a}) \), and the magnitude can be calculated as \(|\mathbf{a}||\mathbf{b}|\sin(\theta)\), where \(\theta\) is the angle between \(\mathbf{a}\) and \(\mathbf{b}\). Understanding these properties is important for handling vector triple products and similar expressions.

Dot Product

The Dot Product is an operation that takes two vectors and returns a scalar, reflecting the degree to which they align. It’s calculated by multiplying the corresponding components of the two vectors and summing those products.

The formula for the dot product is given by:

• \( \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \)

This is also equivalent to the product of their magnitudes and the cosine of the angle between them:\( \mathbf{a} \cdot \mathbf{b} = |\mathbf{a}||\mathbf{b}|\cos(\theta) \).

A key feature of the dot product, crucial in vector triple products, is the simple computation of zero if the vectors are orthogonal (i.e., perpendicular). This makes the dot product particularly useful in simplifying expressions involving vector products, as seen in determining components of vector identities.

Vector Identity

Vector identities are formulas or equations that hold true for vectors under specific operations like addition, cross product, or dot product. These identities help simplify computations and solve complex vector equations quickly.

One key identity relevant here is the vector triple product identity:

• \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \)

Knowing such identities allows you to convert vector cross products into sum or difference formats using simpler dot products. This makes working with the expressions manageable and reduces chances of computational errors. Vector identities are fundamental tools in physics, engineering, and computer graphics, where vectors frequently describe systems.

Mathematical Proof

Mathematical Proof involves demonstrating that a certain statement or theory is absolutely true through a logical sequence of statements. In vector algebra, proofs often involve verifying identities and properties using known axioms or previously proven statements.

For instance, proving the vector triple product identity \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \) starts with breaking down the operation into simplest elements like cross and dot products as defined.

Using foundational properties like distributive laws of vector addition and anticommutative nature of cross products, mathematicians validate these identities. Proving these expressions ensures that they apply universally and can reliably be used in calculations across different applications, reaffirming their conceptual correctness.