One of the most common ways to determine whether two vectors can be combined or not is by multiplying them which is also called the product of two vectors. This process of getting a product between two vectors is called a cross-product of vectors. Wondering how the vector product of two vectors can be found out and what are the techniques used in finding it out? Well, now you can refer to the Vector Product of Two Vectors - Calculation, Examples, Properties, and FAQ article provided by Vedantu for your reference that will help you understand the basics as well as prepare for your exams.
A vector is used to locate a point in space concerning another and is a quantity having magnitude and direction. In a geometrical representation, it can be pictured as an arrow or a line segment with the direction. Here, the length of an arrow indicates the magnitude of this vector.
When multiplying two vectors, it can be done using two methods.
Dot product or Scalar product
Cross product or Vector product
Here, we will discuss the cross product in detail.
Vector Product of Two Vectors
The cross product or vector product obtained from two vectors in a three-dimensional space is treated as a binary operation and is denoted by x. The resulting product, in this case, is always another vector having the same magnitude and direction.
Let us consider the two quantities vectors a and b.
Two vectors a and b are shown in the picture. To answer what a vector product is, look at the calculations below.
a x b = |a| . |b|. Sin(ф) n
|a| is the length or magnitude of vector a.
|b| is the length or magnitude of vector b.
Θ is the angle between both the vectors b and a.
n is a unit vector perpendicular to both vectors a and b.
As shown in the above picture, if the tail of vectors b and a begins from the origin (0,0,0), then the product of two vectors can be represented as
Cx = ay . bz – az . by
Cy = az . bx – ax . bz
Cz = ax . by – ay . bx
Let us define a vector product by taking an example. Consider a vector a = (2,3,4) and b = (5,6,7). Here, ax = 2, ay = 3, and az = 4. bx = 5, by = 6, bz = 7. Putting these values in the above equation and calculating, we get Cx = -3, Cy = 6, and Cz = -3.
The Direction of Product Vector
While you can define a vector product of two vectors and its magnitude from the above equation, the direction of its product vector can be determined using the rule of right-hand thumb.
According to this right-hand thumb rule, we need to curl the fingers of your right hand from vector a to vector b, and then the thumb is pointed towards the direction of the product vector.
Properties of Vector Product
While the scalar or dot product result of two vectors shows the commutative property, and the cross product is non-commutative.
This means, a x b ≠ b x a. However, from the definition of vector product, an x b = - b x a. This is true because of the change in direction of the product vector.
Similar to a scalar product, this vector product determined from multiplying two vectors also shows a distributive property.
Therefore, a x (b + c) = a x b + a x c
As per the characteristics of the vector product, this calculation of the magnitude value of the vector product equals the area of the parallelogram made by the same two vectors.
You will be able to understand the concepts of vector and cross products intricately by going through our study materials. Now, you can also download our Vedantu app for easier access to these study materials, along with the option of online interactive classes to clear your doubts.
Types of Vectors seen in Physics
There are three types of vectors that are observed in Physics and can be provided as follows:
These vectors include displacement vector, force vector, and momentum.
These vectors are the ones that act along an axis and are hence called the axial vectors. Examples of such vectors are angular velocity, angular acceleration, torque.
Pseudo or inertial vectors:
These are used to create an inertial frame of reference and are hence called pseudo vectors or inertial vectors.