Understanding the Vector Cross Product

The question asks for the value of the vector cross product of two expressions involving vectors: \(\left( {\overrightarrow a - \overrightarrow b } \right)\) and \(\left( {\overrightarrow a + \overrightarrow b } \right)\). The vector cross product is a fundamental operation in vector algebra that results in a new vector perpendicular to both original vectors.

Calculating the Vector Cross Product \(\left( {\overrightarrow a - \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow b } \right)\)

To find the vector cross product \(\left( {\overrightarrow a - \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow b } \right)\), we can use the distributive property of the cross product, similar to how we multiply algebraic expressions. The distributive property states that for vectors \(\overrightarrow u\), \(\overrightarrow v\), and \(\overrightarrow w\):

\(\overrightarrow u \times (\overrightarrow v + \overrightarrow w) = (\overrightarrow u \times \overrightarrow v) + (\overrightarrow u \times \overrightarrow w)\)

and

\((\overrightarrow u + \overrightarrow v) \times \overrightarrow w = (\overrightarrow u \times \overrightarrow w) + (\overrightarrow v \times \overrightarrow w)\)

Applying the distributive property to our expression:

\(\left( {\overrightarrow a - \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow b } \right) = \overrightarrow a \times \left( {\overrightarrow a + \overrightarrow b } \right) - \overrightarrow b \times \left( {\overrightarrow a + \overrightarrow b } \right)\)

Now, distribute \(\overrightarrow a\) and \(\overrightarrow b\) into their respective parentheses:

\(= (\overrightarrow a \times \overrightarrow a) + (\overrightarrow a \times \overrightarrow b) - (\overrightarrow b \times \overrightarrow a) - (\overrightarrow b \times \overrightarrow b)\)

Properties of Vector Cross Product Used

Next, we use some key properties of the vector cross product:

• The cross product of any vector with itself is the zero vector: \(\overrightarrow v \times \overrightarrow v = \overrightarrow 0\). This is because the angle between the vector and itself is 0, and \(\sin(0^\circ) = 0\).
• The cross product is anti-commutative: \(\overrightarrow b \times \overrightarrow a = -(\overrightarrow a \times \overrightarrow b)\). This means changing the order of the vectors changes the direction of the resulting vector.

Using these properties, we can substitute into our expression:

\(\overrightarrow a \times \overrightarrow a = \overrightarrow 0\)

\(\overrightarrow b \times \overrightarrow b = \overrightarrow 0\)

\(\overrightarrow b \times \overrightarrow a = -(\overrightarrow a \times \overrightarrow b)\)

Substitute these back into the expanded expression:

\(= \overrightarrow 0 + (\overrightarrow a \times \overrightarrow b) - (-(\overrightarrow a \times \overrightarrow b)) - \overrightarrow 0\)

Simplifying the Expression

Now, simplify the expression:

\(= (\overrightarrow a \times \overrightarrow b) - (-(\overrightarrow a \times \overrightarrow b))\)

\(= (\overrightarrow a \times \overrightarrow b) + (\overrightarrow a \times \overrightarrow b)\)

Combining the terms, we get:

\(= 2(\overrightarrow a \times \overrightarrow b)\)

Thus, the vector cross product \(\left( {\overrightarrow a - \overrightarrow b } \right) \times \left( {\overrightarrow a + \overrightarrow b } \right)\) simplifies to \(2(\overrightarrow a \times \overrightarrow b)\). This calculation demonstrates the application of the distributive property and the properties involving the cross product of identical vectors and the anti-commutative nature of the vector cross product using vectors a and b.

Understanding the properties of cross product is crucial for solving such vector algebra problems involving vectors a and b.