Consider the two vectors $\vec{A}$ and $\vec{B}$ . The sum of their vectors ie: $|\vec{A}+\vec{B}|$ , if $|\vec{A}|>|\vec{B}|$
1) is equal to $|\vec{A}|+|\vec{B}|$
2)must be less than $|\vec{A}|+|\vec{B}|$
3)cannot be greater than $|\vec{A}|+|\vec{B}|$
4)must be equal to $|\vec{A}|-|\vec{B}|$
Ok so i initially got my answer as 3) as sum of vectors is less than or equal to $|\vec{A}|+|\vec{B}|$ but the answer key shows that the answer is 2) and 3)
But I don't understand why it must be less than $|\vec{A}|+|\vec{B}|$
We have many examples where angle is $0°$ where adition of vectors is equal to $|\vec{A}|+|\vec{B}|$ .
Please suggest your explanations .
To answer your question -
Consider the "triangle inequality",
$$||\mathbf{A+B}|| \le ||\mathbf{A}||+||\mathbf{B}||$$
This relation can be proven by using the law of cosines
$$||\mathbf{A+B}||^2=||\mathbf{A}||^2+||\mathbf{B}||^2-2||\mathbf{A}||.||\mathbf{B}||\cos(\mathbf{A,B}),$$
where $$0\le(\mathbf{A,B})\le\pi$$
For proof, see this link