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. What is the vector product of two vectors and how is it calculated in Physics?
The vector product, also known as the cross product, of two vectors results in a third vector perpendicular to both original vectors. It is calculated using the formula: a × b = |a| × |b| × sin(θ) n̂, where |a| and |b| are the magnitudes of the vectors, θ is the angle between them, and n̂ is a unit vector perpendicular to both.
2. How can you determine the direction of a vector product using the right-hand thumb rule?
To find the direction of the vector product, point your right-hand fingers from the first vector to the second (a to b). The direction in which your thumb points is the direction of the resulting vector, following the right-hand thumb rule.
3. In what real-world situations is the vector product commonly applied in Physics?
The vector product is widely used in Physics for:
- Finding torque on a rotating body
- Determining angular momentum
- Calculating the magnetic force on a moving charge (F = q (v × B))
- Finding the area of a parallelogram formed by two vectors
4. What distinguishes the cross product from the dot product of two vectors?
The cross product yields a vector and depends on the sine of the angle between vectors, while the dot product yields a scalar and uses the cosine. The cross product is non-commutative (a × b ≠ b × a), whereas the dot product is commutative.
5. Why is the magnitude of the vector product equal to the area of the parallelogram formed by two vectors?
The magnitude |a × b| represents the area because it is calculated as |a| × |b| × sin(θ), which mathematically equals the area formula for a parallelogram spanned by vectors a and b.
6. Can the vector product of two parallel vectors ever be nonzero?
No, the cross product of two parallel vectors is always zero because the angle between them is either 0° or 180°, and sin(0°) = sin(180°) = 0.
7. How does the non-commutative property of the cross product affect calculations in Physics?
Swapping the order of vectors in a cross product reverses the direction of the resulting vector: a × b = –(b × a). This is essential when calculating torque, magnetic force, and other vector physical quantities where direction matters.
8. What mistakes do students commonly make when solving vector product problems in board exams?
Typical mistakes include:
- Forgetting to use sine instead of cosine for the angle
- Not correctly identifying the direction using the right-hand rule
- Assuming the cross product is commutative
- Mixing up vector and scalar products in calculations
9. How can the cross product be used to find the volume of a parallelepiped?
The volume of a parallelepiped formed by vectors a, b, and c is given by the scalar triple product: |a ⋅ (b × c)|. Here, b × c gives a vector perpendicular to both, and taking the dot product with a gives the volume.
10. Why is understanding vector product essential for mastering advanced Physics topics in CBSE Class 12?
Mastering the vector product is crucial because it forms the foundation for advanced topics like Electromagnetism, Rotational Motion, and Mechanics in Class 12 Physics. Proficiency in cross product calculations ensures accuracy in solving real-world Physics problems as per CBSE syllabus 2025-26.
