Problem 22 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The result is the zero vector, \( \mathbf{0} \).

Step 1: Recognize the Problem Type

We need to evaluate the result of a cross product involving unit vectors. Specifically, the vector triple product \( \mathbf{i} \times (\mathbf{j} \times \mathbf{k}) \).

Step 2: Apply the Vector Triple Product Identity

The vector triple product identity states that \( \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \). Here, \( \mathbf{a} = \mathbf{i} \), \( \mathbf{b} = \mathbf{j} \), and \( \mathbf{c} = \mathbf{k} \).

Step 3: Calculate Dot Products

Calculate \( \mathbf{i} \cdot \mathbf{k} \) and \( \mathbf{i} \cdot \mathbf{j} \). Since \( \mathbf{i} \) is orthogonal to both \( \mathbf{j} \) and \( \mathbf{k} \), both dot products are 0: \( \mathbf{i} \cdot \mathbf{k} = 0 \) and \( \mathbf{i} \cdot \mathbf{j} = 0 \).

Step 4: Substitute into Vector Triple Product Identity

Substituting into the identity, we have: \( (\mathbf{i} \cdot \mathbf{k}) \mathbf{j} - (\mathbf{i} \cdot \mathbf{j}) \mathbf{k} = 0 \cdot \mathbf{j} - 0 \cdot \mathbf{k} = \mathbf{0} \).

Step 5: Draw the Conclusion

The result of the operation \( \mathbf{i} \times (\mathbf{j} \times \mathbf{k}) \) is the zero vector, \( \mathbf{0} \).

Key Concepts

Vector Triple Product Identity

When dealing with cross products that involve three vectors, the vector triple product identity is incredibly useful. The identity is given by: \[ \mathbf{a} \times (\mathbf{b} \times \mathbf{c}) = (\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c} \]This formula allows us to simplify the expression of nested cross products. It expresses the result as a combination of dot products and individual vectors from the original set, making the calculation more manageable.

• \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are the vectors involved.
• The dot products, \(\mathbf{a}\cdot\mathbf{b}\) and \(\mathbf{a}\cdot\mathbf{c}\), determine the coefficients for \(\mathbf{b}\) and \(\mathbf{c}\).

In the given exercise, using \(\mathbf{a} = \mathbf{i}\), \(\mathbf{b} = \mathbf{j}\), and \(\mathbf{c} = \mathbf{k}\), the identity simplifies because the dot products result in zero, leading the whole expression to be the zero vector \(\mathbf{0}\). This type of problem often arises in physics and engineering contexts, where it's important to simplify complex expressions efficiently.

Unit Vectors

Unit vectors are vectors with a magnitude of one. Commonly represented by \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) in three-dimensional space, these vectors help define directions along the x, y, and z axes, respectively.
Why are unit vectors important?

• They help in specifying directions in 3D space.
• In calculations, they simplify operations such as dot and cross products.

In vector notation:

• \(\mathbf{i} = \langle 1, 0, 0 \rangle\) represents the x-axis direction.
• \(\mathbf{j} = \langle 0, 1, 0 \rangle\) stands for the y-axis direction.
• \(\mathbf{k} = \langle 0, 0, 1 \rangle\) indicates the z-axis direction.

For the exercise, understanding unit vectors is crucial because they are involved in the cross product operation that leads to using the vector triple product identity. They are not just abstract mathematical entities but serve as concise building blocks for more complex vector analysis.

Orthogonal Vectors

Orthogonal vectors are vectors that are at right angles to each other. In simpler terms, two vectors are orthogonal if their dot product is zero. This property plays a significant role in vector mathematics and applications.
Consider unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\):

• \(\mathbf{i}\) is orthogonal to both \(\mathbf{j}\) and \(\mathbf{k}\).
• \(\mathbf{j}\) is orthogonal to both \(\mathbf{i}\) and \(\mathbf{k}\).
• \(\mathbf{k}\) is orthogonal to both \(\mathbf{i}\) and \(\mathbf{j}\).

Why does orthogonality matter?

• It simplifies mathematical calculation, especially when dealing with dot products.
• Orthogonality is a foundation for concepts such as vector projections and coordinate transformations.

In the context of the exercise, orthogonality between \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) led to zero dot products, simplifying the application of the vector triple product identity and resulting in a zero vector outcome.