Dot Product, Cross Product, or Resultant force 🤔
No complicated equation or examples
If hope you remember 🅅🄴🄲🅃🄾🅁, a straight line ➡️ with pointing head. And you use it to know the movement of something in particular 🧭direction.
Resultant Force
If 2 people pull something with the same force in completely opposite direction then the movement would be 0. I hope they don’t break it.
But if they show some team work and pull it in the same direction then 𝚛𝚎𝚜𝚞𝚕𝚝𝚊𝚗𝚝 𝚏𝚘𝚛𝚌𝚎 would be higher and they require comparatively less effort.
Resultant vector is nothing but the i͟m͟p͟a͟c͟t͟ o͟f͟ o͟n͟e͟ v͟e͟c͟t͟o͟r͟ o͟n͟ o͟t͟h͟e͟r͟. The result is somewhere between both vectors reclined towards the higher vector.
*𝕋&ℂ 𝕒𝕡𝕡𝕝𝕚𝕖𝕕
Resultant vector is easy to understand because we face it everywhere in daily life. Eg when you pull something, resistance and the gravity make your job difficult. When you drive against the wind you spend more fuel.
Multiplication means combination. Dot product and cross products both are a kind of multiplication.
- Dot product is the interactions between similar dimensions
- Cross product is the interactions between different dimensions
🤓 Do you know, Dot product of two vectors result in a number whereas ❌ Cross product results in another vector. That’s why they are also called Scalar Product and Vector Product respectively.
Dot Product
If you multiply 2 lists, dot product is the multiplication of similar items. Like [🌶️ 🥦 🍍]·[🌶️ 🥦 🍍] = chilly with chilly, Broccoli with Broccoli and Pineapple with Pineapple. It can be written as following for 2 vectors a,b.
Notice that it is the combination of similar components and result in a number.
When we add vectors, it tell us the accumulated growth of all the vectors. But, when we do dot product, it tells us the d̲i̲r̲e̲c̲t̲i̲o̲n̲a̲l̲ ̲g̲r̲o̲w̲t̲h̲ ̲o̲f̲ ̲o̲n̲e̲ ̲v̲e̲c̲t̲o̲r̲ ̲t̲o̲ ̲a̲n̲o̲t̲h̲e̲r̲.
And since chilly with chilly, doesn’t matter which growth comes first.
(3🌶️, b1) ⋅ (4🌶️, b2) = (4🌶️, b2) ⋅ (3🌶️, b1)
But remember if both vectors go in the exact same direction then the growth would be maximum, just like 2 friends riding the bicycle together. Not necessary they’re going on the same bicycle but going in same direction can make the journey comfortable. If one is standing like a right angle then the overall growth would be 0.
Measure of how similar both f̶r̶i̶e̶n̶d̶s̶ vectors are.
You can calculate it by the angle between both vectors. This is an alternative way of calculating the dot product.
𝐚⋅𝐛=‖𝐚‖ ‖𝐛‖𝐜𝐨𝐬(θ);
- |a| is the magnitude (length) of vector a
- |b| is the magnitude (length) of vector b
- θ is the angle between a and b
With the little help of trigonometry, the projection of one vector on other can be calculated and then multiplied by the magnitude of other vector to find their overall growth ie Dot Product.
- If both friend is gonna take the journey together then the angle between them is 0. In other words cos(θ) = 0°.
- If one of the friend is standing like right angle then no journey. Means cos(θ) = 90°.
It doesn’t matter if you take small or big angle between both vectors. Because cos(θ) = cos(360 °— θ)
Applications of Dot product
Do you remember the first picture of this article? A swimmer says “I,ll Dot with it” . Because
Dot product is used when Two forces A and B produce maximum effect or impact when work together ‘in a line’.
So you can use Dot product to determine the minimum angle when 2 vectors can produce maximum force. Eg If you go along with the stream flow in the river 🌊, you’ll have to put less force. There are many other examples like when you drop a ball or throw it straight, or when you fly a plane against the wind. Let’s understand with an example of aligning solar panel for maximum efficiency.
In above picture, vector B is a normal unit vector denoting the direction of solar panel. And vector A is another normal vector pointing to the light source. When there minimum angle or 0 between both vectors, the efficiency would be high.
Resultant vector tells you the direction and strength of result force. But Dot product tells the similarity.
Cross Product
As we read earlier, Cross product is the interactions between different dimensions. Different dimensions help to discover a new dimension.
This newly discovered dimension is perpendicular on both participating dimensions. And this is how you can calculate the coordinates of new dimension.
Remember this winning position of the Tic-Tac-Toe game. It will help you to remember how Dot product and Cross products can be calculated.
If you make a parallelogram from participating vectors, the area of parallelogram is equal to the size(magnitude) of new vector.
Remember the story of 2 cats and 1 monkey!!
If you’re aligned with each other, Monkey is nowhere. Even if their perspectives are opposite but if they don’t leave the path then Monkey is nowhere.
In this picture, cats are the participating vectors of different kind and the monkey is the outcome vector. This is what exactly happens when we change the angle between participating vectors.
- When the angle is 0° or 180°, in other words, when both vectors are on the same path irrespective of their direction then cross product is zero. This is just opposite of dot product.
Cross product looks like a ⚔️ 𝘷𝘪𝘭𝘭𝘢𝘪𝘯 ⚔️. It is stronger when the original vectors have no similarity.
Let’s take the same example of crossing the river 🌊. This time you don’t dot with the river. But you cross it straight like a right angle.
In above picture, vector A represents your movement. Vector B is the the velocity of the water. If the velocity of the water is high, you will probably flow with river. But we’re not talking about resultant vector here. We’re are talking about the ⚔️ 𝘷𝘪𝘭𝘭𝘢𝘪𝘯 ⚔️ vector. So what do you think what will stop you to cross the river? Is it the velocity? No! it is the force caused by the velocity. Yes, friction. It is an outcome caused my your movement and river’s velocity.
Another way to calculate cross product is
𝐚×𝐛=‖𝐚‖ ‖𝐛‖ 𝐬𝐢𝐧(θ) 𝐧;
- |a| is the magnitude (length) of vector a
- |b| is the magnitude (length) of vector b
- θ is the 🅂🄼🄰🄻🄻 angle between a and b
- n is the unit vector at right angles to both a and b
I couldn’t find the best example to explain cross product alone. You may find examples like cork screw opener, or a screw. When we rotate them they go into the object. But this is not only the cross product but due to the threads. Imagine a nail instead of screw and apply the same rotational movement, it’ll not work. So in general, cross product is described when rotations are used like angular momentum, torque, curl, magnetic force etc.
If you find any good word problem related to only vectors cross product then please comment.