Question

two vectors have the same magnitude, what is the range of magnitudes of their possible sum?​

Answer 1

Explanation:

The problem states:

|A¯+B¯|=|A¯|−|B¯|

Assuming |A¯|>|B¯|. And asks us to find the angle between |A¯|and |B¯|. Let’s square both sides:

|A¯+B¯|2=|A¯|2+|B¯|2−2|A¯||B¯|

Clearly, the left hand side of this equation is just the dot product of A¯+B¯ with itself, so lets work that out:

|A¯+B¯|2=(A¯+B¯)⋅(A¯+B¯)=|A¯|2+|B¯|2+2A¯⋅B¯

Setting the right hand side of these two equations equal, we have:

|A¯|2+|B¯|2−2|A¯||B¯|=|A¯|2+|B¯|2+2A¯⋅B¯

⟹−|A¯||B¯|=A¯⋅B¯

Now, recall that A¯⋅B¯=|A¯||B¯|cosθ, where θ is the angle between the two vectors. So, our equation becomes:

−|A¯||B¯|=|A¯||B¯|cosθ

⟹cosθ=−1

⟹θ=π

So our two vectors are parallel, and pointing in opposite directions.