1. ## Vectors

Could anyone help me to understand this question? I keep thinking that |u + v| = |u|+|v| in this scenario

2. Originally Posted by samstark
Could anyone help me to understand this question? I keep thinking that |u + v| = |u|+|v| in this scenario
COME ON!
That is not a question. It is a simple statement which is wrong.
What is the question?
What don't you understand?

3. Originally Posted by Plato
COME ON!
That is not a question. It is a simple statement which is wrong.
What is the question?
What don't you understand?
Sorry, the image I tried to post did not appear the first time. I edited my first post. I do not understand how the magnitude of the sum of the two vectors be less than the magnitude of each vector added up.

4. Originally Posted by samstark
Sorry, the image I tried to post did not appear the first time. I edited my first post.
Well for any two vectors, $\|\vec{U}+\vec{V}\|\le\|\vec{U}\|+\|\vec{V}\|$
That is the triangle inequality. That is a basic axiom for vectors.
The sum of the lengths of two sides of a triangle is greater than the length of the third side.

5. $|\overrightarrow{u} + \overrightarrow{v}|$ Here you add the vectors together keeping the angle, so draw them from head(u) to toe(v) and look at them. If you did it correctly it should look like a roof or a ^ with a bigger angle. Then, draw a line from the start point of u to the end point of v. The new line is your $|\overrightarrow{u} + \overrightarrow{v}|$ It should now look like a triangle

In $|\overrightarrow{u}| + |\overrightarrow{v}|$ you're just adding the total magnitude of the two vectors, so just draw the magnitude of vector v, and then continuing in a straight line draw the magnitude of vector u. Just add the vectors together in a straight line and call it $|\overrightarrow{u}| + |\overrightarrow{v}|$. Done correctly this would just look like a straight line.

Look at what you drew, analyze it, and understand it's an exercise to get you to see why the triangle identity works. They just want you to draw it so that you can also conceptualize it in your mind. The idea is you'll learn the math language more intuitively and it's an opportunity to see why math works the way it does. Good luck!

6. The most fundamental concept here is "a straight line is the shortest distance between two points". The direct line from the base of u to the tip of v is a single straight line which is thus shorter than going from the base of u to the tip of u and then to the tip of v.