Ever wondered what is cross product used for? Or, when or how to use cross-product in real life? If yes, then, you are at the right place at the very right time. The cross product, also known as vector product, is a mathematical operation that takes two vectors and returns a third vector that is perpendicular to both of the original vectors.
The cross-product has numerous applications in various fields, including physics, engineering, and computer graphics. In this exclusive article, we will explore some of the exclusive applications that we see in our day-to-day life. So, without wasting any more time, let’s dive right in…!!!
Applications of Cross Product in Real Life
- Calculating the Moment of a Force
- Computing the Torque on an Object
- Finding Angular Momentum of a Rotating Object
- Calculating the area of a parallelogram
- Robotics and Computer Graphics
- Finding the Axis of Rotation of a Rigid Body
Calculating the Moment of a Force
The very first one in my list of top 6 applications of cross product in real life is that you can calculate the moment of a force by using cross products of two vectors. The moment of a force about a point is nothing but a way to analyze the tendency of the force to cause rotation about that given point.
Mathematically, the moment of a force about the given point (let’s say P) is given by:
M = r x F
where,
r = position of a vector from the point of rotation to the point of applications of the force,
F = Force
M = moment of force
Computing the Torque on an Object
One of the exclusive use of cross product is in computing the torque of an object. A torque is nothing but a measure of the rotational force applied to an object.
So, when you apply force to an object at a certain point, you can compute the torque as the cross product of the force vector and the position vector of the pivot point where the force is applied.
Finding Angular Momentum of a Rotating Object
You can easily find out the angular momentum of a rotating object using the cross products of two vectors. By definition, the angular momentum of a rotating object is the property of a rotating body given by moment of inertia times angular velocity.
Mathematically, you can find out the angular momentum of a rotating body as a cross product of an object’s moment of inertia tensor and its angular velocity.
Calculating the area of a parallelogram
You can use the magnitude of the cross-product of two vectors to calculate the area of a parallelogram by these two vectors. As per the definition, the magnitude of the cross-product of two vectors is equal to the area of the parallelogram formed by those vectors.
Mathematically, the area A of a parallelogram formed by two vectors u and v is given by:
A = |u x v|
where,
A = area of a parallelogram
u and v = parallelogram formed by two vector
|u x v| = magnitude of the cross product of u and v
Robotics and Computer Graphics
Yup, you heard it right. You can use a cross-product of two vectors to calculate the torque required to move a robotic arm. The torque vector is the cross product of the joint angle vector and the joint axis vector.
Similarly, you can also use cross-products in computer graphics to calculate the surface normal vector of a 3D object. The surface normal vector is the cross-product of two vectors that lie on the surface of the object.
Finding the Axis of Rotation of a Rigid Body
Last but not least one in my list of top 6 applications of cross product is that you can find the axis of rotation of a rigid body by using cross products of two vectors.
The axis of rotation of a rigid body can be computed as the eigenvector corresponding to the eigenvalue of the moment of inertia tensor that is the smallest in magnitude.
The moment of inertia tensor can be computed using the cross product of the position vector of each point in the rigid body and its mass.
Some Other Uses of Cross Product in Real Life
Apart from the above-mentioned ones, I am also mentioning a few here.
- Computing the magnetic moment of a current loop
- Finding the direction of a magnetic field
- Determining if two vectors are parallel or perpendicular
- Calculating the vorticity in fluid dynamics
- Calculating the magnetic field vector, etc.
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