Hi, friends I hope you are all well and doing the best in your fields. Today we will discuss the cross or the vector product. In the previous article, we discussed vector quantities, scalar quantities, and the scalar or dot product with their properties, and applications in different fields of science now we can talk about the cross or vector quantities in detail because vectors are used in mathematics, physics, engineering or many other fields. Algebraic operations can also be solved by using the vectors. Vectors are widely used because they can provide the magnitude and direction of a quantity.
The vector product is also known or termed as the cross product. Vector product or in the cross-product are binary vectors or these vectors are perpendicular to each other in the three-dimensional plane. Generally, the cross or the vector product can solve complex algebraic operations like torque, magnetic force, and angular momentum. The mathematics which is the field of science, the cross or the vector product can usually represent the product of the given area with the direction where the two vectors are placed in the Euclidean space or the three-dimensional Cartesian plane. The scalar or the dot product can be represented by the sign or symbol () but the cross or the vector product can be represented by the symbol which is termed a cross. The scalar or the dot product is different from the cross or the vector product because the scalar product can be also termed or used for calculating the projection between two vectors. But the vector or cross product is used for the two perpendicular vectors calculation. Now we can start our deep discussion about the cross or the dot product, algebraic operations, applications, and examples.
In the late 18th century, the Quaternion algebraic operation and the first products of the vectors which are violets the commutative law can be described by the scientist William Rowan Hamilton. The experiment can be performed by William in which he can do the product of two vectors and these are the quaternions and the other part for the product is zero which is scalar then their results also contain the vector or the scalar part. The part of scalar and the vector in the result of William product expressed the cross product of the two vectors which can be represented by the A B and the dot product of the two vectors can be expressed as the A B.
After this, the scientist Josiah Willard Gibbs in the 18th century 1881, with Oliver Heaviside represented the expression that can be used for the dot products of the two vectors and also for the cross product of the two vectors which are given there:
The dot product can be expressed through and written as;
A B
The cross product of the two vectors can be expressed through the cross and written as:
A B
As we can see the expression for both the dot and the cross product expressed that the vector A can be multiplied by the vector B and they can’t violet the commutative law so that's why their matrix can be always 3 3 and it can also be explained by the scientist, Saru's and their law or rule can be termed as Sarrus rule which is given there:
Cross or the vector product can be defined as:
“when the product of the two vectors is the vector quantity it can be represented as A B then it is teremed as the vector product or also the cross product. And the resultant vector which can be denoted by the C are perpendicular to the both of the vector A and the vector B.”
Mathematically the cross or the vector product can be written as:
A B = AB sinθ n
There,
A represented the vector A
B represented the vector B
And,
A represents the magnitude of the vector A
B represented the magnitude of the vector B
The θ represented the angle between the vector A and the vector B which lies in the 0° to 180°. And the unit vector which is perpendicular to the vector A and the vector B can be denoted through n.
The product of the two vectors, vector A and vector B is zero (0) when both of these vectors A and vector B are parallel to each other.
The magnitude and the direction of the vectors can be represented through the right-hand rule. In which the direction can be shown in the right-hand rule and the magnitude of two vector products is always equal to the parallelogram which is given or in which the vector product can be done.
The right-hand rule in the term of the cross or the vector product can be defined as:
"The thumb of the right hand determines the direction of the resultant vector C which is the product of two vectors cross product and when we can curl our finger in the direction of the thumb it indicates the direction the vector A and after proper curling of fingers, it indicates the direction of the vector B."
As we discuss the right-hand rule in terms of the cross or dot product the thumb and the curling finger represent the directions of the vector and also the direction of the resultant vector through the thumb.
In the given figure the cross or vector product of two vectors can be shown. The thumb represents the resultant vector which is equal to the product of two vectors A and the vector B
The fingers and the curl fingers can represent the direction of both vector's magnitude and the θ represents the angle between both of these vectors in the area of a parallelogram.
The product of the two vectors with their units vector, coordinate equation, or the mathematical expression are given there:
Let's suppose the two vectors, the vector A and the vector B which is equal to the,
A = A1 i + A2j + A3k
B = B1 i + B2j + B3k
As we know i , j and k are the unit vectors that can be oriented in the orthonormal basis positively and they can be written as:
k i = j
j k = i
i j = k
Now according to the anti-commutativity law, or when these unit vectors can be oriented negatively then the orthonormal basis can be written as:
i k = – j
j i = – k
k j = – i
Now when we can do the product of the two vectors, vector A and vector B with their unit vector which can follow the distributive law then it can be written as:
A = A1 i + A2j + A3k
B = B1 i + B2j + B3k
And
A B = (A1 i + A2j + A3k) (B1 i + B2j + B3k)
Then,
A B = A1 B1 ( i i ) + A1 B2 ( i j ) + A1 B3 ( i k ) + A2B1 ( j i ) + A2B2 ( j j ) + A2B3 ( j + k ) + A3B1 ( k i ) + A3B2 ( k j ) + A3B3 ( k k) …… (i) equation
We also know that:
i i = j j = k k = 0
Because the vectors are perpendicular and they can't follow the law of the commutative.
By putting the values of the unit vectors in the equation (i)
A B = A1 B1 (0) + A1 B2 (k) – A1 B3 ( j) – A2B1 ( k ) + A2B2(0) +A2B3 (i) + A3B1 ( j) – A3B2 (i) + A3B3 (0)
Then arrange them and then it can be written as;
A B = A2B3 (i) – A3B2 (i) – A1 B3 ( j) + A3B1 ( j) + A1 B2 (k ) – A2B1 ( k )
Now we take common the same unit vectors i, j, and k and write as,
A B = ( A2B3 – A3B2) i + ( A3B1 – A1 B3) j + ( A1 B2– A2B1 ) k
The products of the two vectors, vector A and vector B are the component of the scalar and their resultant vector are C which are equal and written as:
C = C1i + C2j + C3k
So that's why the resultant vectors with their unit vector are equal and written as:
C1i = A2B3 – A3B2
C2j = A3B1 – A1 B3
C3k = A1 B2– A2B1
Also, it can be written in the matrix, column matrix which is given there,
C1i C2j C3k |
A2B3 – A3B2 A3B1 – A1 B3 A1 B2– A2B1 |
=
To represent the vector products the determinants can be used and they can be written as:
i A1 B1 |
j A2 B2 |
k A3 B3 |
A B =
But if we can use the Sarrus rule in the matrix then it can be written as:
A B =( A2B3 (i) + A3B1 j + A1 B2k ) – ( A3B2 i + A1 B3 j + A2B1 k )
Then it can also be written as:
A B = ( A2B3 – A3B2) i + ( A3B1 – A1 B3) j + ( A1 B2– A2B1 ) k
And these are the components of the cross or the vector products.
The characteristics and the main properties of the cross or the scalar product are given there:
Area of a parallelogram
Perpendicular vectors
Self vector product
Violation of the commutative law
Parallel vectors
Anti parallel vectors
Vector product in the rectangular component
Distributivity
Scalar multiplication
Orthogonality
Zero vector
Their detail is given there:
The product of the two vector quantities, the magnitude of these vector A and vector B is equal to the area of a parallelogram along with their sides. The area of the parallelogram is equal to,
Area of a parllelogram= length height
Area of the parallelogram = ( A) (B sinθ)
there,
A represents the length
B represents the height
sinθ represents the angle between vector A and vector B
The total area of the parallelogram with their sides is the magnitude of these vector products. and it can be written as:
Area of a parallelogram = ( A B magniytude)
Also written as:
Area of a parallelogram = A B
When the two vectors, vector A and vector B are perpendicular to each other then their magnitude is always maximum because the angle θ between them is equal to 90°, then it can be written as:
A B = AB sin 90° n
As we know that:
sin 90° = 0
Then,
A B = AB (1) n
A B = AB n
And it is the maximum magnitude of the two vectors in the cross or vector product. But in the case of their unit vectors, it can also be written as:
k i = j
j k = i
i j = k
It can also be written as:
i k = – j
j i = – k
k j = – i
A A = AA sin 0° n
A A = 0° n
A A = 0 this is the zero or the null vector. Another example for the vector B is given there:
B B = BB sin 0° n
B B = 0° n
Then,
B B = 0
For the unit vectors, the self-product is also equal to the null or zero vector and written as:
i i = ii sin 0° n
i i = (1) (1) sin0° n
i i = 0° n
i i = 0
So the other unit vector self-product is also equal to the null or zero vector and can be written as:
j j = 0
k k = 0
And
i i = j j = k k
The cross or the vector product of the two vectors, vector A and the vector B are not to be commutative because they can't follow this law because the vectors are perpendicular to each other.
Let the vector A and the vector B and the sin θ be the angle that is present between the product of these two vectors and it can be written as:
A B = AB sinθ n ……… (i) equation
And if we can reverse them and write them as:
B A = BA sinθ(- n )
And also it can be written as;
B A = AB sinθ(- n ) ……… (ii) equation
According to the commutative law
AB = BA
Now compare the equation (i) and the (ii) equation
A B = – B A
A B ≠ – B A
Thus, it can proved that the cross or the vector product of the two vectors A and b can't follow the commutative property.
When the two vectors, vector, and vector B are parallel to each other then always their cross or the vector product is equal to the null or the zero vector.
Mathematical expressions for parallel vectors are given there:
A B = AB sinθ n
In parallel vectors θ = 0° then,
A B = AB sin (0) n
A B = AB (0) n
A B = (0) n
A B = 0
When the two vectors, vector, and vector B are anti-parallel to each other then always their cross or the vector product is equal to the null or the zero vector.
Mathematical expressions for parallel vectors are given there:
A B = AB sinθ n
In parallel vectors θ =180° then,
A B = AB sin (180) n
A B = AB (0) n
A B = (0) n
A B = 0
The cross or the vector product can be done in the cartesian or the rectangular components and there three components are the scalar but their product result is always the vector quantity.
Let's suppose the two vectors, the vector A and the vector B which is equal to the,
A = A1 i + A2j + A3k
B = B1 i + B2j + B3k
As we know i , j and k are the unit vectors that can be oriented in the orthonormal basis positively and they can be written as:
k i = j
j k = i
i j = k
Now according to the anti-commutativity law, or when these unit vectors can be oriented negatively then the orthonormal basis can be written as:
i k = – j
j i = – k
k j = – i
Now when we can do the product of the two vectors, vector A and vector B with their unit vector which can follow the distributive law then it can be written as:
A = A1 i + A2j + A3k
B = B1 i + B2j + B3k
And
A B = (A1 i + A2j + A3k) (B1 i + B2j + B3k)
Then,
A B = A1 B1 ( i i ) + A1 B2 ( i j ) + A1 B3 ( i k ) + A2B1 ( j i ) + A2B2 ( j j ) + A2B3 ( j + k ) + A3B1 ( k i ) + A3B2 ( k j ) + A3B3 ( k k) …… (i) equation
We also know that:
i i = j j = k k = 0
Because the vectors are perpendicular and they can't follow the law of the commutative.
By putting the values of the unit vectors in the equation (i)
A B = A1 B1 (0) + A1 B2 (k) – A1 B3 ( j) – A2B1 ( k ) + A2B2(0) +A2B3 (i) + A3B1 ( j) – A3B2 (i) + A3B3 (0)
Then arrange them and then it can be written as;
A B = A2B3 (i) – A3B2 (i) – A1 B3 ( j) + A3B1 ( j) + A1 B2 (k ) – A2B1 ( k )
Now we take common the same unit vectors i, j, and k and write as,
A B = ( A2B3 – A3B2) i + ( A3B1 – A1 B3) j + ( A1 B2– A2B1 ) k
The products of the two vectors, vector A and vector B are the component of the scalar and their resultant vector are C which are equal and written as:
C = C1i + C2j + C3k
So that's why the resultant vectors with their unit vector are equal and written as:
C1i = A2B3 – A3B2
C2j = A3B1 – A1 B3
C3k = A1 B2– A2B1
Also, it can be written in the matrix, column matrix which is given there,
C1i C2j C3k |
A2B3 – A3B2 A3B1 – A1 B3 A1 B2– A2B1 |
=
The formula which can be derived from the cross or the vector product can also be written in the form of a determinant and their mathematical expression is given there:
i A1 B1 |
j A2 B2 |
k A3 B3 |
A B =
The cross or the vector product of the two vectors follows the distributivity property. Their mathematical expression is given there:
A ( B + C ) = A B + A C
In cross or the vector product this distributivity property can be proved by the vectors.
The cross or the vector product of the two vectors A and vector B ( A B ) is always orthogonal to the vector A and the vector B.
The scalar multiplication can also be done with the cross or the vector product of the vectors. Their mathematical expression can be written there:
( cA) B = c ( A B)
c represented the scalar multiplication, A represents the vector A, and B represents the vector B.
In the various fields of science, vector or cross products can be used generally but in mathematics, computer graphics, physics, or engineering mostly cross or vector products can be used. Some applications of vector products with details are given there:
Computer graphics
Physics
Engineering
In computer graphics, wide cross or vector products can be used in different programs. The major parts in which the cross or the vector products are used are given there:
Rotations: in the graphics where the algorithm can be used the cross or the vector product is widely used. It can also be used to compute the angular velocity and also to determine the axis of the rotation. In animations or the different stimulating systems cross or the vector product can be used to simply them.
Normal vectors: for the lightening in the calculations in the computer graphing program the normal vectors are used generally. The non-parallel vectors which are lying in the programming of the vector or the cross product can be used to simplify them.
In physics, the cross or the vector product is widely used to solve complex algebraic operations along with geometry the main fields in which the vector product can be used are given there:
Angular momentum: angular momentum is the product of two different vector quantities, one is linear momentum which is denoted by ρ and the other is position vector which can be denoted by the r . Their formula or mathematical expression is given there:
L = p r
Their L denotes the angular momentum.
Angular momentum can be widely used in dynamic rotation or isolated systems.
Torque: torque is the product of two different vector quantities, one is force which is denoted by F and the other is position vector which can be denoted by the r . Their formula or mathematical expression is given there:
τ = F r
Their τ denotes the torque.
Application of vector products in engineering fields where mainly the cross product are used is given there:
Magnetic force: (B)
Moment of a force:F
With time or with the complexity of the quantities or algebraic operations cross or vector products can be used in many different new fields or they can also be improved the advanced topic mainly in which the cross or the dot product can be used are given there:
To simplify the complex vector problem or the complex problem in physics the triple product of the vectors can be used because it can simplify them in a very efficient or accurate way. The mathematical expression or the formula that can be used in vector triple product is given there:
A ( B C ) = ( A . C ) B – ( A . B) C
there,
A represented the vector A
B represented the vector B.
C represented the vector C.
Three vectors can be used in this product so that is why it can also be termed as the triple vector product.
Angular momentum: angular momentum is the product of two different vector quantities, one is linear momentum which is denoted by ρ and the other is position vector which can be denoted by the r . Their formula or mathematical expression is given there:
L = p r
Their L denotes the angular momentum.
Torque: torque is the product of two different vector quantities, one is force which is denoted by F and the other is position vector which can be denoted by the r . Their formula or mathematical expression is given there:
τ = F r
Their τ denotes the torque.
Force of a moving charge: force in the magnetic field that can apply on the charging particle is the product of the two vectors and they are the velocity of the charged particle and the other vector is the magnetic field. their mathematical expression is given there:
F = q ( v B)
There,
F denote the force of the charged particles in the magnetic field.
q denotes the charge of the particles
v denotes the velocity of the charged particles.
B denotes the magnetic field.
Like the scalar or the dot product the cross or the scalar product can play a very vital role in different fields of science and simplify complex quantities or solve complex algebraic problems in engineering, physics, and mainly in mathematics. Cross or the vector product can show the expressing relation between the algebra and the geometric calculations and solve the problems in a very efficient way in Euclidean space. After understanding the applications and the depth of the cross or the vector product, easily the complex problem can be simplified efficiently. With time the cross or vector product is more commonly used in various fields of science.
How close are we to seeing fully autonomous vehicles dominate the roads in the United States? The idea of self-driving cars has captivated the public imagination for years, promising a future where technology handles the complexities of driving, reducing accidents, and easing traffic congestion.
However, despite rapid advancements in automated driving technologies, the reality of a driverless future remains uncertain. Let’s take a look at the current state of autonomous vehicles, the challenges that still need to be overcome, and how far we are from actually realizing the dream of fully automated driving on a large scale.
Automated driving in the United States includes varying levels of automation, from basic driver-assistance systems to highly automated vehicles. The Society of Automotive Engineers (SAE) defines six levels of driving automation , from Level 0 (no automation) to Level 5 (full automation).
Most commercially available vehicles today operate at Level 2, which includes systems like Tesla’s Autopilot and General Motors’ Super Cruise, capable of managing steering, acceleration, and braking under certain conditions but still requiring active supervision by the driver. As of 2024, fully autonomous vehicles (Level 5) remain in the experimental stage, with companies like Waymo and Cruise testing driverless cars in limited urban environments.
Despite progress, the deployment of fully autonomous vehicles is still in its infancy. Widespread adoption of Level 4 or Level 5 vehicles may not occur for many years, due to ongoing technical and regulatory challenges. The regulatory environment varies significantly across states, with some, like California and Arizona, leading in allowing testing and deployment of autonomous vehicles, while others remain more restrictive.
Automated driving has the potential to significantly enhance traffic flow and reduce congestion on U.S. roadways. The U.S. Department of Transportation suggests that widespread adoption of autonomous vehicles could lead to a reduction in traffic congestion by optimizing vehicle spacing, speed, and lane usage.
Autonomous vehicles can communicate with each other and with traffic infrastructure, allowing for smoother transitions and fewer bottlenecks during peak hours. This capability is especially promising in urban areas, where congestion is a persistent problem, costing the U.S. economy billions annually in lost productivity.
Safety improvements are another critical benefit, with the potential to dramatically reduce road accidents caused by human error, which accounts for most crashes, says an Indiana-based semi truck accident lawyer . Autonomous vehicles are designed to eliminate common risky behaviors like distracted driving, speeding, and impaired driving.
Automated braking systems alone could prevent or mitigate a significant percentage of rear-end collisions. By minimizing human error, automated driving systems could potentially save thousands of lives each year, significantly reducing the number of annual fatalities on U.S. roads.
One of the primary technological hurdles facing the development of fully autonomous vehicles is the limitation of current sensor systems. Technologies like LiDAR, radar, and cameras are crucial for detecting and interpreting the driving environment, but each has its limitations. For example, LiDAR provides high-resolution 3D mapping but struggles in poor weather conditions like fog or heavy rain.
Radar can detect objects in various weather conditions but lacks the resolution to differentiate between closely spaced objects. Cameras, while essential for visual recognition, are heavily reliant on good lighting conditions and are prone to misinterpreting shadows or reflections as obstacles. These limitations can lead to dangerous situations, as sensors may fail to accurately perceive or respond to complex driving scenarios.
The current state of AI and machine learning also poses significant barriers. Autonomous vehicles rely on AI to make split-second decisions based on vast amounts of data collected from their sensors.
However, AI systems are not yet advanced enough to handle the full range of unpredictable and nuanced situations that can arise on the road. According to a 2023 study by MIT , current AI models struggle in scenarios where human judgment and experience are crucial, such as interpreting the intentions of pedestrians or reacting to erratic behavior by other drivers.
To pave the way for the safe and effective deployment of autonomous vehicles, several critical areas require substantial change. First, the regulatory framework governing autonomous vehicles needs to be more comprehensive and consistent across states.
Currently, the regulatory landscape is fragmented, with each state setting its own rules for testing and deployment. This lack of uniformity creates challenges for manufacturers and slows progress. Only 29 states have enacted legislation specifically addressing autonomous vehicles so far.
Another key area for advancement is the gradual improvement of driver-assistance systems as a stepping stone toward full autonomy. Enhancing existing technologies, such as advanced driver-assistance systems (ADAS), will help build public trust and gradually introduce more sophisticated levels of automation.
Public education initiatives are also crucial to help consumers understand the capabilities and limitations of these technologies, reducing the likelihood of misuse and increasing acceptance.
Expert predictions on the timeline for fully autonomous vehicles vary widely, reflecting the complexity and uncertainty surrounding the technology's development. Some optimistic projections, like those from Tesla CEO Elon Musk, suggest that fully autonomous vehicles could be widely available by the late 2020s.
However, most industry experts and researchers take a more cautious view. For instance, a 2023 report by BCG predicts that Level 4 autonomy—where vehicles can handle most driving tasks independently in specific environments—might not become mainstream until the late 2030s. The report cites ongoing technological, regulatory, and infrastructural challenges as key reasons for the slower rollout.
The adoption of fully autonomous vehicles is expected to occur unevenly, with urban areas likely seeing these technologies sooner than rural regions. Dense urban environments, where the benefits of reduced congestion and enhanced safety are most significant, are likely to be the initial focus for autonomous vehicle deployment.
These factors suggest that while progress is being made, the widespread presence of fully autonomous vehicles on American roads is still a decade or more away.
Are we truly on the brink of a fully autonomous driving revolution, or is it still a distant goal? The answer lies somewhere in between. While significant progress has been made in developing automated driving technologies, the path to widespread adoption is fraught with challenges, including technological limitations, regulatory hurdles, and public skepticism.
The next decade will likely see incremental advancements, particularly in urban areas, as the industry continues to refine and improve these systems. However, achieving true, widespread autonomy will require coordinated efforts across technology, regulation, and infrastructure, ensuring that safety and reliability are at the forefront of this transformative journey.