Introduction
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
Where,
|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
Example
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
Commutative Property
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.
Distributive Property
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:
Proper vectors:
These vectors include displacement vector, force vector, and momentum.
Axial vectors:
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.
FAQs on Vector Product of Two Vectors
1. Is a x b Equal to b x a, where they are Two Vectors?
The magnitude of a x b and b x a is the same; however, their direction differs. So, effectively a x b = - b x a.
2. Write down the Formula for a Cross Product or Vector Product?
The formula determining the cross product of two vectors is a x b = |a| . |b| . Sin(Ꝋ) n, where |b| and |a| are values of vector b and a respectively and Ꝋ is the angle formed between them.
3. Which Rule is Used to Determine the Direction of a Vector Product?
The right-hand thumb rule states, when you curl your right-hand fingers from the direction of vector a to vector b, then your thumb indicates the direction of the resulting vector.
4. What are the applications of Vector Products?
There is multiple application of vector product and some of them can be provided as follows:
Location of a vector perpendicular to two given vectors. Here the vectors that are provided must not be equal to zero and must also be not equal to zero. The cross product of the two original vectors should be equal to the product of their magnitude or length
Using the product of two vectors to find out the area of a parallelogram.
Using the product of two vectors to find the volume of a parallelepiped. This is a solid which has six sides which are parallelograms.
5. What are the objectives of learning about the Vector Product of Two Vectors?
There are multiple objectives of learning about the Vector Product of Two Vectors and can be provided as follows:
It helps students to understand what is the difference between the scalar product and the vector product that is obtained by the multiplication of two vectors.
It helps to determine the product between two vectors and hence check whether they can be combined or not.
Students also get to understand how vectors are quite useful in Physics and a lot of their calculations.
Vedantu makes sure to fulfil all these objectives by providing notes that help students to understand the working of vectors and their product. Students can also access the Vedantu NCERT solutions for Physics to understand the concepts better.
6. What is a vector and why is it necessary to understand a Vector Product?
A vector is an object that has both the presence of a magnitude and a direction. Geometrically this can be pictured as a line that has been directed to a particular direction whose length will be the magnitude of the vector and the arrow is the one that indicates its direction. The direction of a vector is always read from tail to head. If two vectors have both the magnitude and the direction the same they can be easily multiplied to get the product of these two vectors. The product of vectors helps us with several applications. Physical quantities that are expressed in vector form can be used to find the product and check its compatibility.
7. What is the right-handed screw rule and what is it applied to the product of Vectors?
The right-hand screw rule is used to understand the direction of the vector product. To determine this direction the following steps are used:
The screw is first kept perpendicular to the direction of the plane that will contain both the vectors
Now turn the screw from vector 1 to vector 2. The direction will now be determined as the screw progresses further.
For example, if the screw is moving upwards then the cross product of the vectors will be the vector that is pointing upwards and if the screw is moving downwards then the cross product of the two vectors will be pointing downwards.
8. What is the major difference between the dot product of vectors and cross product of vectors?
The major difference that is seen between the dot product and the cross product is that the dot product is defined as the product of the magnitude of the vectors and the cos of the angle that is present between these two vectors whose product is being found out. While on the other hand it is seen that the cross product will be the product of the magnitude of the two vectors multiplied by the sine of the angle that is present between the two vectors whose product is being found out.
