SOLUTION: Here is my problem, concerning vectors and norms(?) I have the following information: u and v are two vectors. || u || = 3, || v || = sqrt(5), and u·v = 1 [dot product]. With th

Algebra ->  College  -> Linear Algebra -> SOLUTION: Here is my problem, concerning vectors and norms(?) I have the following information: u and v are two vectors. || u || = 3, || v || = sqrt(5), and u·v = 1 [dot product]. With th      Log On


   

Question 939326: Here is my problem, concerning vectors and norms(?)
I have the following information:
u and v are two vectors.
|| u || = 3, || v || = sqrt(5), and u·v = 1 [dot product]. With this information, find
|| u + v || = ?
I know from proofs that it is impossible for || u + v || to be larger than ||u|| + ||v||, but I cannot seem to get the answer for the life of me.
The answer (since this was a practice problem) is 4, and I'd love to know the "how".
Thanks!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Hint: Draw the two vectors where their tails start at the same point. The first vector u is in black while the other vector v is in blue. The angle between the two vectors is shown in purple the angle alpha. Use the parallelogram rule to help you construct the red resultant vector u%2Bv



You can use the formula cos%28alpha%29+=+%28u%2Av%29%2F%28abs%28u%29%2Aabs%28v%29%29 to find the angle alpha. Note: u and v are vectors, so when I say u%2Av I mean "dot product of u and v".

We're dealing with a parallelogram. So the adjacent angles are supplementary meaning that alpha+%2B+theta+=+180. Use the value of alpha to find theta.

Once you know theta, you can then use the law of cosines c%5E2+=+a%5E2%2Bb%5E2+-2ab%2Acos%28C%29 to find the length of the red resultant vector which will be your answer.