Question

Evaluate the cross product which \(\left( {5\widehat I + 3\widehat J} \right) \times \left( { - 4\widehat I + 2\widehat J} \right),\)expands to\( - 20\widehat I \times \widehat I + 10\widehat I \times \widehat J - 12\widehat J \times \widehat I + 6\widehat J \times \widehat J\).

Short answer

The cross product is -\( - 20\left( {\widehat I \times \widehat I} \right) + 10\left( {5\widehat I \times 2\widehat J} \right) + 3\widehat J \times \left( { - 4\widehat I} \right) + \left( {3\widehat J \times 2\widehat J} \right)\)

Step-by-step solution

Given data

Given is the cross product\(\left( {5\widehat I + 3\widehat J} \right) \times \left( { - 4\widehat I + 2\widehat J} \right)\)

Definition of cross product

The cross product is a binary operation on two vectors in three-dimensional space. It creates a vector that is perpendicular to both vectors. The vector product of two vectors, a and b, is represented by a, b. It generates a perpendicular vector to both a and b. Vector items are also known as cross goods. The cross product of two vectors produces a vector that may be calculated using the Right-hand Rule.

If \(\overrightarrow A \)and \(\overrightarrow B \) lie in the \(xy\) plane, we can use the results for the unit vectors to calculate the cross product, which will be in the \( + z\)or \( - z\) direction

\(\begin{aligned}{}\overrightarrow A \times \overrightarrow B &= ({A_x}\widehat i + {A_y}\widehat j) \times ({B_x}\widehat i + {B_y}\widehat j)\\ &= \left( {{A_x}{B_x}} \right)\widehat i \times \widehat j + \left( {{A_y}{B_y}} \right)\widehat j \times \widehat j + \left( {{A_x}{B_y}} \right)\widehat i \times \widehat j + \left( {{A_y}{B_x}} \right)\widehat j \times \widehat i\end{aligned}\)

Evaluate the given cross product

The given cross product can be evaluated as below:

\(\begin{aligned}{}\left( {5\widehat I + 3\widehat J} \right) \times \left( { - 4\widehat I + 2\widehat J} \right) &= 5\widehat I \times \left( { - 4\widehat I + 2\widehat J} \right) + 3\widehat J \times \left( { - 4\widehat I + 2\widehat J} \right)\\& = \left( {5\widehat I \times \left( { - 4\widehat I} \right)} \right) + \left( {5\widehat I \times 2\widehat J} \right) + 3\widehat J \times \left( { - 4\widehat I} \right) + \left( {3\widehat J \times 2\widehat J} \right)\\ &= - 20\left( {\widehat I \times \widehat I} \right) + 10\left( {5\widehat I \times 2\widehat J} \right) + 3\widehat J \times \left( { - 4\widehat I} \right) + \left( {3\widehat J \times 2\widehat J} \right)\end{aligned}\)

Hence, the product is \( - 20\left( {\widehat I \times \widehat I} \right) + 10\left( {5\widehat I \times 2\widehat J} \right) + 3\widehat J \times \left( { - 4\widehat I} \right) + \left( {3\widehat J \times 2\widehat J} \right)\)