When Does Vector Addition Equal the Sum of Magnitudes?

[antiflag403]

Hey everyone,
I realize that this is a pretty simple problem but i can't seem to wrap my brain around it. If someone could point me in the right direction i would be thankful.
suppose (vector)C= (vector)A + (vector)B
a) under what circumstances does [(vector)C]= [(vector)A] + [(vector)B]? ( [ ]= absolute value)
b) could [(vector)C]= [(vector)A] - [(Vector)B]? if so how? if not, why not?
ok. for A i was thinking the only way that could be true is if both A and B had the same sign, but I am pretty sure that's wrong.
For B I don't think both C=A+B and [C]=[A]- could be true, but I am not sure why.
If someone could guide me in the right direction i would be grateful. THANKS!

 

[quantumdude]

I think that you'll go a long way in understanding this if you try out some examples.

Why don't you try to check your equalities with the following:

1.) $\vec{A}=3\hat{i}+4\hat{j}$, $\vec{B}=9\hat{i}+12\hat{j}$
2.) $\vec{A}=3\hat{i}+4\hat{j}$, $\vec{B}=5\hat{i}+13\hat{i}$
3.) $\vec{A}=3\hat{i}+4\hat{j}$, $\vec{B}=-3\hat{i}-4\hat{j}$
4.) $\vec{A}=3\hat{i}+4\hat{j}$, $\vec{B}=0\hat{i}+0\hat{j}$

edited to add:

Have you been taught the dot product of two vectors?

 

[Architeuthis Dux]

Hello there,
It seems like you are struggling with understanding vector addition and subtraction. Let me try to explain it in simpler terms for you.

Firstly, let's define what a vector is. A vector is a mathematical quantity that has both magnitude and direction. It is usually represented by an arrow pointing in a specific direction.

Now, let's look at vector addition. When we add two vectors, we are essentially combining their magnitudes and directions. This means that the resulting vector will have a magnitude equal to the sum of the magnitudes of the two vectors and a direction that is determined by the angle between the two vectors. So, if vector A has a magnitude of 5 and vector B has a magnitude of 3, the resulting vector C will have a magnitude of 8 (5+3) and a direction determined by the angle between A and B.

To answer your first question, [(vector)C]= [(vector)A] + [(vector)B] is true when the two vectors are added in the same direction. This means that the angle between A and B is 0 degrees, making the direction of C the same as A and B.

Now, let's move on to vector subtraction. When we subtract one vector from another, we are essentially finding the difference between their magnitudes and directions. This means that the resulting vector will have a magnitude equal to the difference between the magnitudes of the two vectors and a direction determined by the angle between the two vectors. So, if vector A has a magnitude of 5 and vector B has a magnitude of 3, the resulting vector C will have a magnitude of 2 (5-3) and a direction determined by the angle between A and B.

To answer your second question, [(vector)C]= [(vector)A] - [(Vector)B] is possible if the two vectors are subtracted in the same direction. This means that the angle between A and B is 0 degrees, making the direction of C the same as A and B.

I hope this helps to clear things up for you. Remember, vectors are all about magnitude and direction, so make sure to pay attention to those when performing vector operations. Good luck!