Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 11: Problem 21
Show that \((a) \hat{\mathbf{i}} \times \hat{\mathbf{i}}=\hat{\mathbf{j}} \times \hat{\mathbf{j}}=\hat{\mathbf{k}} \times \hat{\mathbf{k}}=0\) (b) \(\hat{\mathbf{i}} \times \hat{\mathbf{j}}=\hat{\mathbf{k}}\) \(\hat{\mathbf{i}} \times \hat{\mathbf{k}}=-\hat{\mathbf{j}},\) and \(\hat{\mathbf{j}} \times\hat{\mathbf{k}}=\hat{\mathbf{i}}\)
Short Answer
Expert verified
(a) Zero since vectors are parallel. (b) Use the right-hand rule for direction.
Step by step solution
01
Understand Cross Product Property
The cross product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\), denoted \(\mathbf{A} \times \mathbf{B}\), is a vector that is perpendicular to both \(\mathbf{A}\) and \(\mathbf{B}\). The magnitude of \(\mathbf{A} \times \mathbf{B}\) is given by \(|\mathbf{A}||\mathbf{B}|\sin\theta\), where \(\theta\) is the angle between \(\mathbf{A}\) and \(\mathbf{B}\). If \(\mathbf{A}\) and \(\mathbf{B}\) are parallel (or identical), the angle \(\theta = 0\), making \(\sin\theta = 0\) and thus the cross product is zero.
02
Part (a) Solution
Given that the unit vectors \(\hat{\mathbf{i}}\), \(\hat{\mathbf{j}}\), and \(\hat{\mathbf{k}}\) are orthogonal unit vectors directed along the axes of a right-handed coordinate system. If we take the cross product of any unit vector with itself, such as \(\hat{\mathbf{i}} \times \hat{\mathbf{i}}\), the vectors are parallel, hence \(\hat{\mathbf{i}} \times \hat{\mathbf{i}} = 0\). Similarly, \(\hat{\mathbf{j}} \times \hat{\mathbf{j}} = 0\) and \(\hat{\mathbf{k}} \times \hat{\mathbf{k}} = 0\). Therefore, \[\hat{\mathbf{i}} \times \hat{\mathbf{i}} = \hat{\mathbf{j}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}} \times \hat{\mathbf{k}} = 0.\]
03
Part (b.i) Cross Product of \(\hat{\mathbf{i}}\) and \(\hat{\mathbf{j}}\)
The cross product \(\hat{\mathbf{i}} \times \hat{\mathbf{j}}\) follows the right-hand rule: if the fingers of the right hand curve from \(\hat{\mathbf{i}}\) to \(\hat{\mathbf{j}}\), the thumb points in the direction of the cross product, which is \(\hat{\mathbf{k}}\). Thus, \(\hat{\mathbf{i}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}}\).
04
Part (b.ii) Cross Product of \(\hat{\mathbf{i}}\) and \(\hat{\mathbf{k}}\)
Using the right-hand rule, if the fingers curl from \(\hat{\mathbf{i}}\) to \(\hat{\mathbf{k}}\), the thumb points in the reverse direction of \(\hat{\mathbf{j}}\), hence \(\hat{\mathbf{i}} \times \hat{\mathbf{k}} = -\hat{\mathbf{j}}\).
05
Part (b.iii) Cross Product of \(\hat{\mathbf{j}}\) and \(\hat{\mathbf{k}}\)
Following the right-hand rule again, if the fingers curve from \(\hat{\mathbf{j}}\) to \(\hat{\mathbf{k}}\), the thumb points in the direction of \(\hat{\mathbf{i}}\). Therefore, \(\hat{\mathbf{j}} \times \hat{\mathbf{k}} = \hat{\mathbf{i}}\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Vector Multiplication
Vector multiplication, often referred to as the cross product, is a mathematical operation performed on two vectors in three-dimensional space. The result of this operation is another vector that is orthogonal, or perpendicular, to both of the original vectors.
- The cross product of vectors \(\mathbf{A}\) and \(\mathbf{B}\) is denoted as \(\mathbf{A} \times \mathbf{B}\).
- The direction of the resulting vector is determined by the orientation of the original vectors in space.
- The magnitude of the cross product is calculated using the formula \(|\mathbf{A}||\mathbf{B}|\sin\theta\), where \(\theta\) is the angle between \(\mathbf{A}\) and \(\mathbf{B}\).
Right-Hand Rule
The right-hand rule is a simple and effective tool to determine the direction of the cross product vector. To apply it, imagine using your right hand:
- Align your fingers in the direction of the first vector, \(\mathbf{A}\).
- Then, curl your fingers towards the second vector, \(\mathbf{B}\).
- Your thumb will point in the direction of the resultant vector of the cross product, \(\mathbf{A} \times \mathbf{B}\).
- \(\hat{\mathbf{i}} \times \hat{\mathbf{j}} = \hat{\mathbf{k}}\) - Curl from \(\hat{\mathbf{i}}\) to \(\hat{\mathbf{j}}\), thumb points in direction of \(\hat{\mathbf{k}}\).
- \(\hat{\mathbf{i}} \times \hat{\mathbf{k}} = -\hat{\mathbf{j}}\) - Curl from \(\hat{\mathbf{i}}\) to \(\hat{\mathbf{k}}\), thumb opposes \(\hat{\mathbf{j}}\).
- \(\hat{\mathbf{j}} \times \hat{\mathbf{k}} = \hat{\mathbf{i}}\) - Curl from \(\hat{\mathbf{j}}\) to \(\hat{\mathbf{k}}\), thumb aligns with \(\hat{\mathbf{i}}\).
Unit Vectors
Unit vectors are vectors that have a magnitude of one and are typically used to indicate direction. In three-dimensional space, the coordinate axes have standard unit vectors:
- \(\hat{\mathbf{i}}\) for the x-axis,
- \(\hat{\mathbf{j}}\) for the y-axis,
- \(\hat{\mathbf{k}}\) for the z-axis.
- Any unit vector crossed with itself results in a vector of zero magnitude -> \(\hat{\mathbf{i}} \times \hat{\mathbf{i}} = 0\).
- The cross product of different unit vectors follows the rules determined by the coordinate system.
Orthogonal Vectors
Orthogonal vectors are vectors that are at right angles to each other, meaning they have a dot product of zero. In the context of the unit vectors \(\hat{\mathbf{i}}\), \(\hat{\mathbf{j}}\), and \(\hat{\mathbf{k}}\), each pair is orthogonal:
- \(\hat{\mathbf{i}} \cdot \hat{\mathbf{j}} = 0\),
- \(\hat{\mathbf{i}} \cdot \hat{\mathbf{k}} = 0\),
- \(\hat{\mathbf{j}} \cdot \hat{\mathbf{k}} = 0\).
- If two vectors are orthogonal, the cross product produces a non-zero vector.
- When using unit vectors, this guides the expected direction of the resultant vector according to the right-hand rule.
