Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
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  
-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

1 Answer 1

up vote 1 down vote accepted

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.

share|improve this answer
    
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
1  
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.