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## Algebra (all content)

### Course: Algebra (all content)>Unit 19

Lesson 9: Adding vectors in magnitude & direction form

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
• 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.
• if A vector + B vector = A vector - B vector.then find magnitude of B vector.?
• 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.
• 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?
• 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.
• 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.
• The magnitude of the sum, and the sum of the magnitude. This is really confusing me!
• 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
! :)
• 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!
• 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?
• 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?