Find ||u+v|| given ||u||, ||v||, and the dot product of u and v

Question

If ||u|| = 5, ||v|| = 1, and u * v = -3, find ||u+v||

Note that "*" defers to taking the dot product when used between vectors, and multiplication otherwise

How might I go about this? The only thing I can think of is the definition of the dot product, which tells you that u * v = ||u|| * ||v|| * cosx, and therefore if u * v < 0, the angle between u and v is obtuse (since cosx will be greater than 90 degrees). But that doesn't help me solve the problem I don't think

Any help is appreciated!

Answer 1

Recall $\|u + v\|^2 = (u + v \mid u + v) = (u \mid u)+(u \mid v)+(v \mid u)+(v \mid v) = \|u\|^2+2(u \mid v)+\|v\|^2.$ Substitute and use the fact that $\|x\| \geq 0$ to obtain back $\|u + v\|.$

Answer 2

Note that $$ \lVert u+v\rVert^2=(u+v)\cdot(u+v)=\lVert u\rVert^2+\lVert v\rVert^2+2(u\cdot v) $$ by expanding the dot product and using linearity.