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### Course: Precalculus (Eureka Math/EngageNY) > Unit 2

Lesson 5: Topic D: Lessons 20: Vectors and stone bridges- Direction of vectors from components: 1st & 2nd quadrants
- Direction of vectors from components: 3rd & 4th quadrants
- Direction of vectors
- Vector components from magnitude & direction
- Vector components from magnitude & direction
- Adding vectors in magnitude & direction form (1 of 2)
- Add vectors: magnitude & direction to component
- Adding vectors in magnitude & direction form (2 of 2)
- Add vectors: magnitude & direction
- Vector addition & magnitude

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# Vector addition & magnitude

Sal explains some interesting properties of the magnitude of vector sums. Created by Sal Khan.

## Want to join the conversation?

- but according to triangular law of vectors a+b=c.Then how a+bcan be greater than c .this will contradict triangular law of vectors(12 votes)
- It is important to understand that algebraic addition and vector addition are different things. While a+b=c means that c equals a+b algebraically, this is not the case with vectors. We cannot add the magnitudes of two vectors to get the resultant like we would add 2 & 3 to get 5; unless they act in the same direction.(32 votes)

- if A vector + B vector = A vector - B vector.then find magnitude of B vector.?(7 votes)
- subtract A vector from both sides.

Now you have B vector = - B vector

now add B vector to both sides.

Now you have 2 B vector = O vector. Divide by two and you see that B is the zero vector, and thus must have magnitude zero.

In vectors, just like the real numbers, the only vector which is its own opposite is the zero vector.(8 votes)

- In the quiz "Adding vectors in magnitude and direction form", arctangent is sometimes opposite/adjacent and sometimes adjacent/opposite. I'm not sure why this is, it's very confusing. Also, the tips say to add 2pi or pi to some answers while not giving any reasoning. None of the videos that I can find in the Vector section explain why 2pi must be added or why arctangent sometimes needs to slip the opposite and adjacent. Sal briefly mentions "modifications" to arctangents in "Total Displacement While Hiking" but doesn't go in depth...could anyone help me on this?(5 votes)
- Well, has 4 quadrants and the tangent pass in two of then. The calculator just is defined by (-pi) and (pi), that means its just defined for the first and the fourth quadrant. But sometimes u must calculated the tangent of the second or third quadrant in radiussss (or degrees) and to do that and you must sum or subtract pi or 2*pi ( 180º or 360º).

So, you must see whitch quadrant is the result of the sum of your vectors, then you must know that the calculator will give you the anwear of arctangent between pi and -pi (180º and -180º), then sum or subtract pi or 2*pi (180º or 360º) to put it in the way that the question asks you. Recalling that tangent of minus 50 (-50) is Also the tangent of 310.(6 votes)

**Explanation for why ||C||>||A||+||B|| is impossible:**! :)

We can say that "The sum of the two legs of a triangle is always larger than the hypotenuse" but here is an easy way to get an intuitive grasp on it.

Since vector addition forms a triangle, we can think about angles that are formed at the intersections. We know that the larger the angle, the larger the side opposite to it.

So, for the hypotenuse to be larger than the two legs, the angle opposite of the hypotenuse has to be larger than the angle opposite to the two legs.

The sum of the angles inside a triangle must equal 360 degrees. This means that the angle opposite of the hypotenuse must be greater than 180 degrees.

180 degrees is a straight line. If you try to make the angle larger, you WILL form a triangle, but the angle opposite of the hypotenuse will be 180-x with x being the angle that you add.

That is why ||C||>||A||+||B|| is impossible!

Hope this helps someone(4 votes)- Great explanation! Thinking about it in terms of angles is a helpful way to understand why ||C||>||A||+||B|| is impossible. The angle opposite the hypotenuse must be greater than 180 degrees, which is impossible in a triangle. Therefore, ||C||>||A||+||B|| is impossible because it violates the rules of triangles. Thank you for sharing your explanation!(2 votes)

- I'm thinking about a relation to add vectors mathematically directly without graphing the resultant vector.

Instead we would get its components directly and input it into the Pythagorean relation :

V resultant = sqrt( (V1cosΘ+V2)^2 + (V1sinΘ)^2 )

Where Θ is the included angle between the two vectors.(4 votes) - The magnitude of the sum, and the sum of the magnitude. This is really confusing me!(4 votes)
- Wait if Sal moved the direction of vector B, then wouldn't that change both the DIRECTION and MAGNITUDE of vector C? If so, then that means vector C is no longer vector C, right?

So, how come it doesn't matter that vector C is changed?(2 votes) - This is just a question about vectors in general, but are vectors always represented by a straight line? Why? How would you represent a curve using vectors?(0 votes)
- Vectors are just the relative relationship between two locations and so the unique description of that is a straight line. If you want to describe curves you might try using parametric equations. Check out Pikachu for example: http://m.wolframalpha.com/input/?i=pikachu+curve(5 votes)

- Is it only possible for ||a|| + ||b|| = ||c|| if vectors an and b are scalar multiples of each other? Is this equivalent to them being parallel? Finally, does anyone know how to get an arrow above the letters to denote a vector?(2 votes)
- Wait, so these properties are pretty much related to basic triangle properties? I think I'm getting a sort of misconception here.(1 vote)
- No, you're absolutely correct. The rules for triangles apply to vectors too.(2 votes)

## Video transcript

Sal:Let's say that we have three vectors, vectors A, B and C, and we know that vector A plus vector B
is equal to vector C. Now given this I have some
interesting questions. Can you construct a scenario where the magnitude of vector C is equal to the magnitude of vector A plus
the magnitude of vector B? And can you also, using potentially
different vectors A and B, construct a scenario where the magnitude of vector C is greater than the magnitude of vector A plus
the magnitude of vector B? I encourage you to pause this video right now and try to do that. Try to come up with some
vectors A and B so that when you take their sum,
the magnitude of the sum is equal to the sum of the magnitudes. And also see if you can come
up with some vectors A and B so that if you take the sum of
the vectors that the magnitude of the sum is actually greater
than the sum of the magnitude. see if you can come up with that. I'm assuming you've given a go at it, and potentially you've gotten
a little bit frustrated, especially with the second one. The only way- Let's actually
just draw some vectors. If you have vector A like this, and let's say vector B
looks something like that, then A plus B ... We can just shift this
over, copy and paste. A plus B is going to look like this. A plus B, or vector C I guess we could say, is going to look like that. And notice, these three
vectors always form a triangle. If you have a triangle, one side cannot be longer than the sum of
the other two sides. Think about it. If you wanted this to
be longer what you could try to do is maybe change
vector B in a way so you're pushing it further and further out. Maybe if you change your
vector B a little bit you could get this vector C
to be longer and longer. Maybe if you made your vector B like this. Maybe your vector B would
look something like this. Now your vector C is getting
pretty long, but it's still shorter than the
sum of these two sides. To make it equal to the sum of these two sides you essentially have to make these two vectors go in the
exact same direction. To make it equal you have to
have vector A looking like this. You need to change the direction of B, or essentially construct a vector B that's going in the exact same direction. Only in this circumstance
will you get this scenario, where the magnitude of
vector C is equal to. Really the largest that
the magnitude of the sum can be is the sum of the
magnitudes, and that only happens when these two are
going in the same direction. These are going in the same,
in the exact same direction. This right over here is impossible. You could never get one side
of a triangle being longer than the sum of the other two sides,
based on what we just saw. You're probably saying, "What
about the circumstance where "the magnitude of our sum is
less than the magnitude of ..." Than the sum of the magnitudes,
I guess I could say. This is fairly typical. This is pretty much always the scenario, this is what's always going to be the case when the vectors are not
in the same direction. If someone drew a vector like this- Let me draw that a little bit straighter. If someone drew a vector
like this and a vector like this these clearly are not
going in the same direction, so the sum of these two vectors, the magnitude of that is going to be less than the sum of these two magnitudes. For example if the
magnitude here is five and the magnitude here is three then we know that if we were to add
these two things ... Let me just show you. Copy and paste. Actually let me just cut and paste, so that we can clean
things up a little bit. Cut and paste. Let's add these two vectors. So we know that the sum of these two, which is going to be this
vector right over here, its magnitude is going to be
less than five plus three. It is going to be less than eight. The only way that this magnitude
could even get to eight is if these two vectors went
in the exact same direction.