The cross product of two vectors is a mathematical operation that results in a vector, perpendicular to the two original vectors we take the cross product of. It is used in various fields, including physics, mathematics, and computer graphics, to understand and deal with vector behaviors in a 3D space.
Suppose that we have two vectors:
Note: Linearly independent vectors cannot be expressed as a linear combination of each other meaning both vectors don't have the same direction.
The cross product can be represented in two ways: geometric and algebraic.
Geometrically, the formula to find the cross product of the two vectors is:
The magnitude of the resultant vector is important in various applications where understanding the area or strength of a vector's effect is important. For instance, in physics, when we find the torque using the cross product, the magnitude helps us understand the strength of our resultant vector's force. We can find the magnitude by calculating the area of a parallelogram that is formed by
When we calculate the cross product of two vectors, our resultant vector has two possible directions: one towards the negative axis and the other towards the positive. This direction cannot be calculated directly through the above formula, so we use the right-hand rule to find it.
We follow the following instructions in the right hand rule trick.
We use our right hand.
We point our index finger in the direction of the first vector,
We point our middle finger in the direction of the second vector,
The resultant vector,
Let's understand the use of the following formula using an example. We can calculate the magnitude of our resultant vector
Let's say
Magnitude of
Magnitude of
Value of
Magnitude of
Note: Our cross product is maximum when the vectors are perpendicular (angle equals
) and minimum when the vectors are parallel (angle equals ).
Algebraically, the cross product between two vectors is calculated using a matrix determinant approach.
Let's suppose we have
The cross product will be:
Here i, j, and k are unit vectors in the x, y, and z directions, respectively.
We get the following formula after simplifying the determinant form:
Let's say
The cross product will be:
Let's discuss the properties of a cross product. Let's suppose
Property name | Explanation | Representation |
Anti-commutative | If we reverse the order of vectors, it results in the opposite sign in the result. | v × w = - w × v |
Distributive | The cross product of a vector with the sum of two other vectors is equal to the sum of the cross products of the vector with each of those two vectors separately. | v × (w + u) = (v × w) + (v × u) |
Zero vector | If any of the vector has the magnitude equal to zero then the resultant cross product is 0. | v × w = 0 if v = 0 or w = 0 |
Jacobi identity | If we take three vectors and perform cross products between them, the sum of all those cross products will be zero. | v × (w × u) +w × (u × v)+u × (v × w) = 0 |
The cross product of vectors is a fundamental mathematical operation with wide-ranging applications in various fields. By taking the cross product of two vectors, we obtain a new vector perpendicular to both the original vectors.
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