Problem 48
Question
If \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0},\) show that $$\mathbf{a} \times \mathbf{b}=\mathbf{b} \times \mathbf{c}=\mathbf{c} \times \mathbf{a}$$
Step-by-Step Solution
Verified Answer
The vectors form a closed loop, resulting in equal cross products.
1Step 1: Understand Vector Addition
We begin by understanding the given condition: \( \mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0} \). This implies that \( \mathbf{a} + \mathbf{b} = -\mathbf{c} \). It tells us that the sum of these three vectors is zero, meaning they form a closed triangle.
2Step 2: Consider Cross Products with Vector Identity
The vector identity for cross product states that for any vectors \( \mathbf{p} \) and \( \mathbf{q} \), \( \mathbf{p} \times \mathbf{q} = - (\mathbf{q} \times \mathbf{p}) \). Using this, we know that cross products are antisymmetric. Also, if \( \mathbf{q} = \mathbf{0} \), \( \mathbf{p} \times \mathbf{q} = \mathbf{0} \).
3Step 3: Express in Terms of Other Cross Products
Substitute \( \mathbf{c} = -(\mathbf{a} + \mathbf{b}) \) into \( \mathbf{b} \times \mathbf{c} \):\[ \mathbf{b} \times \mathbf{c} = \mathbf{b} \times (-(\mathbf{a} + \mathbf{b})) = - (\mathbf{b} \times \mathbf{a} + \mathbf{b} \times \mathbf{b}) \] Because \( \mathbf{b} \times \mathbf{b} \) equals zero, this simplifies to \( -(\mathbf{b} \times \mathbf{a}) \). By antisymmetry, this is equal to \( \mathbf{a} \times \mathbf{b} \).
4Step 4: Show Equality for Remaining Products
Next, we substitute \( \mathbf{b} = -(\mathbf{a} + \mathbf{c}) \) into \( \mathbf{c} \times \mathbf{a} \) to get:\[ \mathbf{c} \times \mathbf{a} = (-(\mathbf{a} + \mathbf{b})) \times \mathbf{a} = - (\mathbf{a} \times \mathbf{a} + \mathbf{b} \times \mathbf{a}) \] With \( \mathbf{a} \times \mathbf{a} = 0 \), this reduces to \((\mathbf{b} \times \mathbf{a}) = \mathbf{a} \times \mathbf{b} \), confirming \( \mathbf{c} \times \mathbf{a} = \mathbf{a} \times \mathbf{b} \).
5Step 5: Confirm Final Equality
For completeness, we also show \( \mathbf{a} = -(\mathbf{b} + \mathbf{c}) \) into \( \mathbf{a} \times \mathbf{b} \): \[ \mathbf{a} \times \mathbf{b} = (-(\mathbf{b} + \mathbf{c})) \times \mathbf{b} = - (\mathbf{b} \times \mathbf{b} + \mathbf{c} \times \mathbf{b}) \] Since \( \mathbf{b} \times \mathbf{b} = 0 \), this confirms \( -(\mathbf{c} \times \mathbf{b}) = \mathbf{c} \times \mathbf{a} \). All equalities hold across \( \mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a} \).
Key Concepts
Cross ProductVector IdentityAntisymmetric PropertyVector Addition
Cross Product
In vector mathematics, the cross product is a crucial operation that helps us determine the vector perpendicular to two given vectors in three-dimensional space. If we have two vectors, say \( \mathbf{a} \) and \( \mathbf{b} \), then the cross product, denoted as \( \mathbf{a} \times \mathbf{b} \) gives us another vector. This resulting vector is perpendicular to the plane containing both \( \mathbf{a} \) and \( \mathbf{b} \).
Importantly, the magnitude of the cross product vector is given by the product of the magnitudes of the original vectors and the sine of the angle \( \theta \) between them, expressed as \( | \mathbf{a} \times \mathbf{b} | = |\mathbf{a}||\mathbf{b}|\sin\theta \).
Here are a few essential properties of the cross product:
Importantly, the magnitude of the cross product vector is given by the product of the magnitudes of the original vectors and the sine of the angle \( \theta \) between them, expressed as \( | \mathbf{a} \times \mathbf{b} | = |\mathbf{a}||\mathbf{b}|\sin\theta \).
Here are a few essential properties of the cross product:
- The cross product is not commutative, meaning \( \mathbf{a} \times \mathbf{b} eq \mathbf{b} \times \mathbf{a} \).
- If vectors \( \mathbf{a} \) and \( \mathbf{b} \) are parallel, then their cross product is zero because the angle between them is zero, so \( \sin\theta = 0 \).
- The direction of the cross product follows the right-hand rule, where if you point your index and middle fingers in the directions of \( \mathbf{a} \) and \( \mathbf{b} \) respectively, your thumb will point in the direction of \( \mathbf{a} \times \mathbf{b} \).
Vector Identity
A vector identity is a formula or equation that is always true for vectors for any given set of vectors. For the cross product, one such essential identity is \( \mathbf{p} \times \mathbf{q} = - (\mathbf{q} \times \mathbf{p}) \). It tells us that switching the order of vectors in a cross product changes the sign of the resulting vector.
In the context of the given exercise, this identity is helpful in rearranging and equating different cross product expressions. Knowing this identity, we can easily simplify and confirm the relationships given in vector problems, such as demonstrating that \( \mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a} \).
Other common vector identities include the triple scalar product identity and those involving vector dots and cross. Recognizing and applying these identities can significantly simplify complex vector calculations.
In the context of the given exercise, this identity is helpful in rearranging and equating different cross product expressions. Knowing this identity, we can easily simplify and confirm the relationships given in vector problems, such as demonstrating that \( \mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a} \).
Other common vector identities include the triple scalar product identity and those involving vector dots and cross. Recognizing and applying these identities can significantly simplify complex vector calculations.
Antisymmetric Property
The antisymmetric property is a key characteristic of the cross product in vector mathematics. This property indicates that for any two vectors \( \mathbf{a} \) and \( \mathbf{b} \), the cross product \( \mathbf{a} \times \mathbf{b} \) satisfies \( \mathbf{a} \times \mathbf{b} = - (\mathbf{b} \times \mathbf{a}) \).
The antisymmetry is evident as switching the order of the vectors results in the negative of the original cross product. This property is widely used not just in solving problems but also in proving vector relationships and equations.
For example, when given that \( \mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0} \), and you need to prove that \( \mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a} \), knowing that \( \mathbf{b} \times \mathbf{c} = - (\mathbf{c} \times \mathbf{b}) \) is dimensional in reaching a solution. Antisymmetry assists in transforming cross product identities into verifiable statements.
The antisymmetry is evident as switching the order of the vectors results in the negative of the original cross product. This property is widely used not just in solving problems but also in proving vector relationships and equations.
For example, when given that \( \mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0} \), and you need to prove that \( \mathbf{a} \times \mathbf{b} = \mathbf{b} \times \mathbf{c} = \mathbf{c} \times \mathbf{a} \), knowing that \( \mathbf{b} \times \mathbf{c} = - (\mathbf{c} \times \mathbf{b}) \) is dimensional in reaching a solution. Antisymmetry assists in transforming cross product identities into verifiable statements.
Vector Addition
Understanding vector addition is essential when dealing with vector equations and expressions. Vector addition involves combining two or more vectors to result in another vector. If you have vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) such that \( \mathbf{a} + \mathbf{b} + \mathbf{c} = \mathbf{0} \), it implies that these vectors form a closed loop, such as a triangle.
This concept can be visualized geometrically, where vector addition translates to moving head to tail along the vectors. In the given exercise, the sum \( \mathbf{a} + \mathbf{b} = - \mathbf{c} \) simplifies how we see and equate the vectors in terms of their cross products.
Here are some properties of vector addition that are important to recognize:
This concept can be visualized geometrically, where vector addition translates to moving head to tail along the vectors. In the given exercise, the sum \( \mathbf{a} + \mathbf{b} = - \mathbf{c} \) simplifies how we see and equate the vectors in terms of their cross products.
Here are some properties of vector addition that are important to recognize:
- It is commutative: \( \mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a} \).
- It is associative: \( (\mathbf{a} + \mathbf{b}) + \mathbf{c} = \mathbf{a} + (\mathbf{b} + \mathbf{c}) \).
- The zero vector \( \mathbf{0} \) acts as an identity element, such that \( \mathbf{a} + \mathbf{0} = \mathbf{a} \).
Other exercises in this chapter
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