What is \(\left( {\vec a - \vec b} \right) \times \left( {\vec a + \vec b} \right)\) equal to?
Understanding Vector Cross Products
The question asks us to simplify the expression \( \left( {\vec a - \vec b} \right) \times \left( {\vec a + \vec b} \right) \), which involves the cross product of two vector expressions. To solve this, we will use the properties of the vector cross product.
Applying the Distributive Property of Cross Product
The cross product is distributive over vector addition and subtraction, similar to how multiplication works with numbers. This means we can expand the given expression:
\( \left( {\vec a - \vec b} \right) \times \left( {\vec a + \vec b} \right) = \vec a \times \left( {\vec a + \vec b} \right) - \vec b \times \left( {\vec a + \vec b} \right) \)
Now, we apply the distributive property again to each term:
\( \vec a \times \left( {\vec a + \vec b} \right) = \vec a \times \vec a + \vec a \times \vec b \)
\( \vec b \times \left( {\vec a + \vec b} \right) = \vec b \times \vec a + \vec b \times \vec b \)
Substituting these back into the main expression:
\( \left( {\vec a \times \vec a + \vec a \times \vec b} \right) - \left( {\vec b \times \vec a + \vec b \times \vec b} \right) \)
Now, we simplify using key properties of the vector cross product.
Key Properties of Vector Cross Product
Here are the essential properties we need:
| Property | Description | Formula |
|---|---|---|
| Cross product of a vector with itself | The cross product of any vector with itself is the zero vector. | \( \vec v \times \vec v = \vec 0 \) |
| Anti-commutativity | The order of vectors in a cross product matters. Swapping the order changes the direction of the resulting vector. | \( \vec b \times \vec a = - \left( \vec a \times \vec b \right) \) |
| Distributivity | Cross product distributes over addition/subtraction. | \( \vec u \times (\vec v + \vec w) = \vec u \times \vec v + \vec u \times \vec w \) |
Simplifying the Expression
Using the properties:
- \( \vec a \times \vec a = \vec 0 \)
- \( \vec b \times \vec b = \vec 0 \)
- \( \vec b \times \vec a = - \left( \vec a \times \vec b \right) \)
Substitute these into the expanded expression:
\( \left( \vec 0 + \vec a \times \vec b \right) - \left( - \left( \vec a \times \vec b \right) + \vec 0 \right) \)
\( = \vec a \times \vec b - \left( - \left( \vec a \times \vec b \right) \right) \)
\( = \vec a \times \vec b + \vec a \times \vec b \)
Combining the like terms:
\( = 2 \left( \vec a \times \vec b \right) \)
Conclusion
The expression \( \left( {\vec a - \vec b} \right) \times \left( {\vec a + \vec b} \right) \) simplifies to \( 2 \left( \vec a \times \vec b \right) \). This result comes directly from applying the distributive property and the property that the cross product of a vector with itself is the zero vector, along with the anti-commutative property.
Therefore, the correct option is \( 2\left( {\vec a \times \vec b} \right) \).
Revision Table: Vector Operations
| Operation | Symbol | Input | Output | Properties (Key) |
|---|---|---|---|---|
| Dot Product | \( \vec a \cdot \vec b \) | Two vectors | Scalar | Commutative (\( \vec a \cdot \vec b = \vec b \cdot \vec a \)), Distributive, \( \vec a \cdot \vec a = |\vec a|^2 \) |
| Cross Product | \( \vec a \times \vec b \) | Two vectors | Vector (perpendicular to both) | Anti-commutative (\( \vec a \times \vec b = - (\vec b \times \vec a) \)), Distributive, \( \vec a \times \vec a = \vec 0 \) |
| Vector Addition | \( \vec a + \vec b \) | Two vectors | Vector | Commutative, Associative |
Additional Information on Vector Cross Product
The magnitude of the cross product \( |\vec a \times \vec b| \) is equal to \( |\vec a| |\vec b| \sin(\theta) \), where \( \theta \) is the angle between \( \vec a \) and \( \vec b \). Geometrically, \( |\vec a \times \vec b| \) represents the area of the parallelogram formed by vectors \( \vec a \) and \( \vec b \). The direction of \( \vec a \times \vec b \) is perpendicular to both \( \vec a \) and \( \vec b \), following the right-hand rule.
The cross product is defined only for three-dimensional vectors. It is widely used in physics and engineering, for example, to calculate torque and magnetic force.
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