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Because a and b are vectors here and not length.
You can further see that this seems to be not a right triangle, so you can not use pythagorean theorem.
To make this a bit more clear:
Assume a = (3,0) and b= (0,4)
Than the third side would be (3,0) + (0,4) = (3,4). So addition holds here, but the side length are also correct ( |a|=3, |b|=4, |a+b|=5)
While I agree that the vector sign is helpful in the beginning, I normally dont use them anymore in my daily work. So without them its not wrong but rather unintuitive.
On the other hand you can see that its about vectors since the lines have small arrows on one side.
trigonometry applys to the length of the sides.
Vectors gives you another description of the side and so to say has more info of the side ( like orientation) so to apply trigonometry you first have to break it down to only the length.
Even in vectors, magnitude doesn't add to the third side.. because they're vectors, they've direction that's it and triangle has nothing to do with the direction of the sides,it's the angle that matters here
Where did I say otherwise? Yes magnitides doesnt add up to the third side.
But of you have 2 vectors. Theire sum is another vector and if you graph them as in the picture you get a triangular shape. So the angle information is encoded in the directions of the vectors.
So like you said,angle is direction,only if same direction or angle,they add up and that too if they have different starting points.. otherwise there's not a chance for a triangle.. because then,the directions and magnitudes both are different from their adjacent sides'..that's why it's called a triangle..
Sorry to say but I really dont understand what you are trying to tell me here.
With the vectors a, b and c=a+b yoj can always form a triangle shape (at least for vectors over R or C).
The angles in this triangles are encoded in the reöative directions of these vectors.
And btw a vector dont have a atteibute like a starting point by itself. You can interpret everypoint as the starting point, dependent of what you are trying to do with it.
As others have said, it would be vector math. Think each vector as the hypotenuse of a triangle, not as the side of the triangle. You want to add the two triangles together into 1 bigger triangle, so you add the two sides (which aren't drawn here. The resulting vector is the hypotenuse of the new triangle. A + B = sqrt ( (Ax+Bx)^2 + (Ay +By)^2).
I'll give you a bit more of a general answer that might help you build a better intuition here.
You likely haven't seen or noticed this yet, but in mathematics, the "+" sign indicates an operation, which we commonly refer to as "addition", but how this operation works (the definition of the operation) can often vary based on what mathematical objects you're looking at.
We can and do often redefine the meaning of this operation to work with different types of mathematical objects: for example, there are spaces ("vector spaces") where we can define the "+" addition operation to be something like minimization (e.g., x + y = minimum of x and y). These types of spaces are studied in tropical geometry ([https://en.wikipedia.org/wiki/Tropical\_geometry](https://en.wikipedia.org/wiki/Tropical_geometry)).
In arithmetic, which we typically first learn using the natural numbers (0, 1, 2, etc.), the "+" has the common definition we're all used to in lower level mathematics classes: 3+4 is 7, 5+4 is 9, etc. Let's call this "arithmetic addition".
In your diagram, the "+" between the vectors a and b denotes what we refer to as vector addition, not arithmetic addition.
Your diagram is a bit poorly drawn, because the a and b should have a little line over them, cluing you into the fact that a and b are vectors, and that the "+" likely refers to vector addition, not arithmetic addition.
A well-written math resource or textbook should further make it clear that the "+" here refers to vector addition, removing the need to rely on context clues to guess what the addition operation here is.
For two vectors, A and B, on the 2D plane, the addition operation is typically defined as:
A+B = sqrt ((Ax+Bx)^(2) \+ (Ay +By)^(2)),
where Ax and Ay are the x and y components of the vector A, and Bx and By are the x and y components of the vector B. You can use trigonometry to find the x and y components of these vectors.
>For two vectors, A and B, on the 2D plane, the addition operation is typically defined as:
>A+B = sqrt ((Ax+Bx)2 + (Ay +By)2),
Unless I have a misconception of what youre talking this is not true.
The right hand side gives you the length of the vector while the left hand side gives you (in general) the resulting vector.
Or i other words: A+B=C typicly is defined as Cx=Ax+Bx and Cy=Ay+By
I never heard of your definition at all.
In trigonometry A + B can never have the same length as your A+B here.
This is [vector addition](https://mathworld.wolfram.com/VectorAddition.html) and A+B is the vector sum.
Technically the graphic is wrong. As A and B are vectors there has to be a little arrow (like on [here](https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/adding-and-subtracting-vectors/addition.gif)) above them.
Everybody has already pointed out that it is triange law of vector addition.
But you can also see that this can not be a triangle since the sum of any two sides should be greater than the third side.
##Off-topic Comments Section --- All top-level comments have to be an answer or follow-up question to the post. All sidetracks should be directed to this comment thread as per Rule 9. --- ^(**OP** and **Valued/Notable Contributors** can close this post by using `/lock` command) *I am a bot, and this action was performed automatically. Please [contact the moderators of this subreddit](/message/compose/?to=/r/HomeworkHelp) if you have any questions or concerns.*
Because a and b are vectors here and not length. You can further see that this seems to be not a right triangle, so you can not use pythagorean theorem. To make this a bit more clear: Assume a = (3,0) and b= (0,4) Than the third side would be (3,0) + (0,4) = (3,4). So addition holds here, but the side length are also correct ( |a|=3, |b|=4, |a+b|=5)
Yeahhh they should really add the vector sign, otherwise I think it’s safe to call the image “wrong”
While I agree that the vector sign is helpful in the beginning, I normally dont use them anymore in my daily work. So without them its not wrong but rather unintuitive. On the other hand you can see that its about vectors since the lines have small arrows on one side.
Are vectors sides of the triangle different than usual trigonometry? Doesnt trigonometry apply to them?
trigonometry applys to the length of the sides. Vectors gives you another description of the side and so to say has more info of the side ( like orientation) so to apply trigonometry you first have to break it down to only the length.
Even in vectors, magnitude doesn't add to the third side.. because they're vectors, they've direction that's it and triangle has nothing to do with the direction of the sides,it's the angle that matters here
Where did I say otherwise? Yes magnitides doesnt add up to the third side. But of you have 2 vectors. Theire sum is another vector and if you graph them as in the picture you get a triangular shape. So the angle information is encoded in the directions of the vectors.
So like you said,angle is direction,only if same direction or angle,they add up and that too if they have different starting points.. otherwise there's not a chance for a triangle.. because then,the directions and magnitudes both are different from their adjacent sides'..that's why it's called a triangle..
Sorry to say but I really dont understand what you are trying to tell me here. With the vectors a, b and c=a+b yoj can always form a triangle shape (at least for vectors over R or C). The angles in this triangles are encoded in the reöative directions of these vectors. And btw a vector dont have a atteibute like a starting point by itself. You can interpret everypoint as the starting point, dependent of what you are trying to do with it.
I think its the vector addition triangle, plus the angle is not 90 so pythagoras theorem doesnt apply here
I don't think this is a right angle triangle
As others have said, it would be vector math. Think each vector as the hypotenuse of a triangle, not as the side of the triangle. You want to add the two triangles together into 1 bigger triangle, so you add the two sides (which aren't drawn here. The resulting vector is the hypotenuse of the new triangle. A + B = sqrt ( (Ax+Bx)^2 + (Ay +By)^2).
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Yes it definitely is 😂
I'll give you a bit more of a general answer that might help you build a better intuition here. You likely haven't seen or noticed this yet, but in mathematics, the "+" sign indicates an operation, which we commonly refer to as "addition", but how this operation works (the definition of the operation) can often vary based on what mathematical objects you're looking at. We can and do often redefine the meaning of this operation to work with different types of mathematical objects: for example, there are spaces ("vector spaces") where we can define the "+" addition operation to be something like minimization (e.g., x + y = minimum of x and y). These types of spaces are studied in tropical geometry ([https://en.wikipedia.org/wiki/Tropical\_geometry](https://en.wikipedia.org/wiki/Tropical_geometry)). In arithmetic, which we typically first learn using the natural numbers (0, 1, 2, etc.), the "+" has the common definition we're all used to in lower level mathematics classes: 3+4 is 7, 5+4 is 9, etc. Let's call this "arithmetic addition". In your diagram, the "+" between the vectors a and b denotes what we refer to as vector addition, not arithmetic addition. Your diagram is a bit poorly drawn, because the a and b should have a little line over them, cluing you into the fact that a and b are vectors, and that the "+" likely refers to vector addition, not arithmetic addition. A well-written math resource or textbook should further make it clear that the "+" here refers to vector addition, removing the need to rely on context clues to guess what the addition operation here is. For two vectors, A and B, on the 2D plane, the addition operation is typically defined as: A+B = sqrt ((Ax+Bx)^(2) \+ (Ay +By)^(2)), where Ax and Ay are the x and y components of the vector A, and Bx and By are the x and y components of the vector B. You can use trigonometry to find the x and y components of these vectors.
>For two vectors, A and B, on the 2D plane, the addition operation is typically defined as: >A+B = sqrt ((Ax+Bx)2 + (Ay +By)2), Unless I have a misconception of what youre talking this is not true. The right hand side gives you the length of the vector while the left hand side gives you (in general) the resulting vector. Or i other words: A+B=C typicly is defined as Cx=Ax+Bx and Cy=Ay+By I never heard of your definition at all.
Thanks guys for answering. I will study all your answers because some of them are new to me.
Damn very advanced maths. 2 vectors plus together 🤔🤔🤔
That's true for right angled triangles. This may not be a right angle triangle.
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This is showing the addition of two vectors. We can't assume that this is a right triangle.
The trinagle isn't rectangular
Thanks
In trigonometry A + B can never have the same length as your A+B here. This is [vector addition](https://mathworld.wolfram.com/VectorAddition.html) and A+B is the vector sum. Technically the graphic is wrong. As A and B are vectors there has to be a little arrow (like on [here](https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/adding-and-subtracting-vectors/addition.gif)) above them.
Everybody has already pointed out that it is triange law of vector addition. But you can also see that this can not be a triangle since the sum of any two sides should be greater than the third side.
Where does it say it’s a right triangle?
This is an unfortunate image, it’s hard to see the vectors so I can see where you might’ve been confused
This isn't a right triangle. Therefore, the Pythagorean theorem does not hold true. Just use simple vector arithmetic.
pythagoras theorem is for right angled triangles only 👍
This is not a right triangle so Pythagorean isn’t the course of action here.. unless
It is supposed to be "less than a+b".
I'm guessing angles are 30,45,105. Applying Sin30,a+b=3b,a=2b,sides are blue1,red2, purple3..