# What are the smallest and largest values of vectors

Let v and w be two vectors with ||v|| = 3 and ||w|| = 4. What is the largest and smallest possible values of v · w?

I found this question in the textbook and I'm not sure how to solve this since I just started linear algebra. Any help is appreciated.

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Hint: $|v\cdot w|\leq \|v\|\|w\|$. – Alex Becker Sep 21 '12 at 3:53
First, find a formula relating $\|v\|,\|w\|,v\cdot w$ and the cosine of the angle between $v$ and $w$. Then, use it. – Gerry Myerson Sep 21 '12 at 3:54
cos = ((v · w)/ ∥v∥ * ∥w∥)? – Jason Curt Sep 21 '12 at 3:58
Yes, that's the one you want. Now, what are the largest and smallest values that a cosine can have? – user22805 Sep 21 '12 at 4:16
-1 and 1, so the dot product of v and w can only be a max of 12 and a min of -12? – Jason Curt Sep 21 '12 at 4:18

We have $||v+w||^2=|| v||^2+ ||w||^2+2(v\cdot w)$. The smallest possible value of $||v+w||^2$ is $1$, when the vectors are pointing in opposite directions, and the largest possible value is $49$, when they point in the same direction.

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Yes but I was looking for the smallest and largest values for v · w to which im pretty sure is -12 and 12 – Jason Curt Sep 21 '12 at 5:06
Indeed. Look above. Smallest value of $||v+w||^2$ is $1$, so smallest value of $2(v\cdot w)$ is $1-9-16=-24$, so smallest value of $v\cdot w$ is $-24/2$. Largest possible value of $||v+w||^2$ is $49$, so largest value of $2(v\cdo w)$ is $49-9-16=24$. largest value of $v\cdot w$ is $24/2$. – André Nicolas Sep 21 '12 at 5:22