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Chapter 10: Problem 53

Use geometry to identify the cross product (do not compute!). $$\mathbf{i} \times(\mathbf{j} \times \mathbf{k})$$

Short Answer

Expert verified
The cross product \(\mathbf{i} \times(\mathbf{j} \times \mathbf{k})\) equals to the zero vector.

Step by step solution

01

Recognize the Vector Relationships

It's crucial to understand that \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) are unit vectors pointing along the x, y, and z axes respectively and are mutually orthogonal to each other. They follow specific relationships such as \(\mathbf{i} \times \mathbf{j} = \mathbf{k}\), \(\mathbf{j} \times \mathbf{k} = \mathbf{i}\), and \(\mathbf{k} \times \mathbf{i} = \mathbf{j}\).
02

Compute the Inner Cross Product

Looking at the problem, we first need to compute the cross product of \(\mathbf{j}\) and \(\mathbf{k}\) inside the parentheses. Using the relationships in Step 1, we know that \(\mathbf{j} \times \mathbf{k} = \mathbf{i}\). Therefore we simplify \(\mathbf{i} \times(\mathbf{j} \times \mathbf{k})\) to \(\mathbf{i} \times \mathbf{i}\).
03

Deal with the Outer Cross Product

Now address the cross product outside of the parentheses: \(\mathbf{i} \times \mathbf{i}\). Because any vector crossed with itself produces a zero vector, the cross product \(\mathbf{i} \times \mathbf{i}=0\).
04

Finalize the Answer

The cross product of \(\mathbf{i}\) and \((\mathbf{j} \times \mathbf{k})\) results in the zero vector.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Vectors
Unit vectors are fundamental in understanding vector operations, as they are vectors with a magnitude of one. They are typically used to represent the directions of the standard axes in space.
For instance, in three-dimensional space:
  • \(\mathbf{i}\) is the unit vector along the x-axis.
  • \(\mathbf{j}\) is the unit vector along the y-axis.
  • \(\mathbf{k}\) is the unit vector along the z-axis.
These unit vectors are mutually orthogonal, meaning each is at a right angle to the other two.
This property of unit vectors makes them incredibly useful in defining directions and performing vector calculations.
Orthogonal Vectors
Orthogonality in vectors means that two vectors are perpendicular to each other.
More formally, if the dot product of two vectors is zero, they are said to be orthogonal.
This is a vital concept in vector operations, particularly in physics and engineering where right angles indicate no influence between directions or forces.Consider the standard unit vectors \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\). Each pair is orthogonal:
  • \(\mathbf{i} \cdot \mathbf{j} = 0\)
  • \(\mathbf{j} \cdot \mathbf{k} = 0\)
  • \(\mathbf{k} \cdot \mathbf{i} = 0\)
These properties are essential when performing vector cross products, as they ensure that the result of such operations is not influenced by components along the axis of any original vector.
Zero Vector
The zero vector is a special type of vector that has a magnitude of zero and thus no direction.
It is represented as \(\mathbf{0}\) and is unique because it remains unchanged when added to any other vector.
In vector cross product operations, if you cross a vector with itself, the result is always the zero vector.For example, in the exercise \(\mathbf{i} \times \mathbf{i} = \mathbf{0}\), which illustrates that crossing identical vectors leads to the zero vector."
The zero vector acts as the identity element in vector addition and a neutral outcome in vector multiplication.
Vector Operations
Vector operations are the backbone of vector calculus and include both vector addition and multiplication.
The cross product is a vector multiplication method that results in another vector. This new vector is orthogonal to the original pair.When calculating a cross product:
  • It uses the right-hand rule to determine the direction.
  • It has a magnitude equal to the area of the parallelogram spanned by the vectors.
In our exercise, this is evident in the calculation \(\mathbf{i} \times (\mathbf{j} \times \mathbf{k})\).
Performing the inner cross product first (\(\mathbf{j} \times \mathbf{k} = \mathbf{i}\)) and then crossing that result with \(\mathbf{i}\) yields the zero vector, illustrating that a vector crossed with itself results in no new vector.

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