If a and b are vectors such that ||a||=7 and ||b||=11, then find the smallest possible value of ||a+b||.
So far I know that for a= $\begin{pmatrix} x \\ y \end{pmatrix}$, $x^2 +y^2 = 49$ and for b= $\begin{pmatrix} m \\ n \end{pmatrix}$, $m^2+n^2 =121$.
What do I do now?
Thanks
Best Answer
Since $||b||=||(a+b)+(-a)||\le||a+b||+||a||$, $\;\;||a+b||\ge||b||-||a||=4$.
This is the smallest value, since $b=-\frac{11}{7}a\implies||a+b||=\frac{4}{7}||a||=4$.