Problem 26 - Advanced Engineering Mathematics

Question

Find the indicated scalar or vector without using \((5),(13)\), or \((15)\). $$ \mathbf{i} \times \mathbf{k}-2(\mathbf{j} \times \mathbf{i}) $$

Step-by-Step Solution

Answer

The result is the vector \(-\mathbf{j} + 2\mathbf{k}\).

Step 1: Understand the Problem

We need to find the result of the vector expression \( \mathbf{i} \times \mathbf{k} - 2(\mathbf{j} \times \mathbf{i}) \). Here, \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the standard unit vectors in three-dimensional space. \( \mathbf{i} \) represents the x-axis, \( \mathbf{j} \) the y-axis, and \( \mathbf{k} \) the z-axis.

Step 2: Compute \( \mathbf{i} \times \mathbf{k} \)

The cross product \( \mathbf{i} \times \mathbf{k} \) results in the vector \( -\mathbf{j} \). This follows the right-hand rule for cross products in three-dimensional space.

Step 3: Compute \( \mathbf{j} \times \mathbf{i} \)

The cross product \( \mathbf{j} \times \mathbf{i} \) results in the vector \( -\mathbf{k} \). This is again using the right-hand rule for cross products.

Step 4: Multiply by Scalar

Now compute \(-2(\mathbf{j} \times \mathbf{i})\). Since \( \mathbf{j} \times \mathbf{i} = -\mathbf{k} \), we multiply by \(-2\): \(-2)(-\mathbf{k}) = 2\mathbf{k}\).

Step 5: Combine Results

Finally, substitute the computed values back into the expression. We have \( \mathbf{i} \times \mathbf{k} = -\mathbf{j} \) and \(-2(\mathbf{j} \times \mathbf{i}) = 2\mathbf{k}\). Therefore, the expression \( \mathbf{i} \times \mathbf{k} - 2(\mathbf{j} \times \mathbf{i}) = -\mathbf{j} + 2\mathbf{k} \). This simplifies to \(-\mathbf{j} + 2\mathbf{k}\), which is a vector.

Key Concepts

Cross Product

The cross product is a fundamental operation in vector calculus used to find a vector that is perpendicular to two given vectors. Unlike the dot product, which results in a scalar, the cross product results in a vector. It is notably used in physics and engineering to determine the torque or rotational effect.
To compute the cross product of two vectors \( \mathbf{A} \) and \( \mathbf{B} \), represented by \( \mathbf{A} \times \mathbf{B} \), follow these rules:

• It is only defined in three-dimensional space.
• It results in a vector perpendicular to both \( \mathbf{A} \) and \( \mathbf{B} \).
• The magnitude of the cross product is given by \( |\mathbf{A} \times \mathbf{B}| = |\mathbf{A}| |\mathbf{B}| \sin(\theta) \), where \( \theta \) is the angle between the vectors.

In the context of unit vectors in three-dimensional space, such as \( \mathbf{i}, \mathbf{j}, \text{ and } \mathbf{k} \), specific cross product results are:

• \( \mathbf{i} \times \mathbf{j} = \mathbf{k} \)
• \( \mathbf{j} \times \mathbf{k} = \mathbf{i} \)
• \( \mathbf{k} \times \mathbf{i} = \mathbf{j} \)
• Notably, \( \mathbf{i} \times \mathbf{k} = -\mathbf{j} \) as seen in the given problem.

Unit Vectors

Unit vectors are vectors with a magnitude of one and they are significant in describing directions in space. In three-dimensional space, the most common unit vectors are \( \mathbf{i}, \mathbf{j}, \text{ and } \mathbf{k} \). Each of these vectors points in the positive direction of the x, y, and z axes respectively.

• \( \mathbf{i} = (1, 0, 0) \) represents the x-axis.
• \( \mathbf{j} = (0, 1, 0) \) represents the y-axis.
• \( \mathbf{k} = (0, 0, 1) \) represents the z-axis.

The concept of unit vectors is crucial in vector calculus as they allow for the easy representation of any vector in terms of basic directions in space. For example, a vector \( \mathbf{A} = a\mathbf{i} + b\mathbf{j} + c\mathbf{k} \) conveys how far the vector "reaches" along each axis.
Unit vectors are essential in simplifying problems involving vector operations, like the cross product, by providing a clear and straightforward directional component to deal with.

Right-Hand Rule

The right-hand rule is a simple tool used to determine the direction of the resultant vector when calculating the cross product. It ensures consistency and accuracy in assessing vector directions in three-dimensional space.
To apply the right-hand rule:

• Align your right hand's thumb with the first vector \( \mathbf{A} \).
• Align your right hand's fingers with the second vector \( \mathbf{B} \).
• Your palm will naturally face in the direction of \( \mathbf{A} \times \mathbf{B} \), indicating the vector perpendicular to both.

In the given solution, for example, using the right-hand rule for \( \mathbf{i} \times \mathbf{k} \) results in \(-\mathbf{j} \) because when you point your thumb along \( \mathbf{i} \) and your fingers along \( \mathbf{k} \), your palm faces in the \(-\mathbf{j} \) direction.
This rule is essential for ensuring that vector directions are calculated correctly in a consistent manner, especially when visualizing and dealing with complex vector problems.

Three-Dimensional Space

Three-dimensional space is an essential concept in mathematics and physics, where points are described using three coordinates: x, y, and z. This space allows us to visualize objects and their interactions in a more realistic context, compared to two-dimensional space.
In mathematical terms, three-dimensional space can be represented using unit vectors \( \mathbf{i}, \mathbf{j}, \text{ and } \mathbf{k} \), each pointing along one of the axes:

• The x-axis is horizontal and runs left to right.
• The y-axis is vertical and runs up and down.
• The z-axis adds depth, running forward and backward.

Each point or vector in this space is represented as an ordered triplet \( (x, y, z) \), simplifying the expression of geometric and physical ideas, such as location, force, and velocity. It also allows for complex operations like vector calculus that involve cross products and dot products.
Three-dimensional space is crucial for accurately describing real-world scenarios, such as the movement of objects, allowing us to apply mathematical principles effectively to other sciences and engineering fields.