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Recently, I do self-learning "Linear Algebra" by using this book "Introduction to Linear Algebra, 3rd Edition" by Gilbert Strang with his lecture on MIT Opencourseware. I am having problem with one of his problem in his book. The following is the problem description.

The figure shows $\frac{1}{2}v + \frac{1}{2}w$. Mark the points $\frac{3}{4}v + \frac{1}{4}w$ and $\frac{1}{4}v + \frac{1}{4}w$ and $v + w$.

enter image description here

For $\frac{1}{4}v + \frac{1}{4}w$ and $v + w$ are very easy. The first one is half of vector u and the second one is double of vector u. The problem is $\frac{3}{4}v + \frac{1}{4}w$.

Here is my attempt:

First, I double the vector u, so $2u = v + w$. Then, I can construct vector v by connecting from vector w to vector 2u. enter image description here

Then, I plot $\frac{1}{4}w$ on the vector w and $\frac{3}{4}v$ on the vector v. Connecting those two point and I get $\frac{3}{4}v + \frac{1}{4}w$. See the picture below. enter image description here

Questions

  • Do I have the correct reasoning?
  • Is there any better way to plot $\frac{3}{4}v + \frac{1}{4}w$?
Monkey D Luffy
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    Alternatively, given $u=\frac12v+\frac12w$, then $\frac34v+\frac14w=\frac12(\frac32v+\frac12w)=\frac12(v+\frac12v+\frac12w)$, which is $\frac12(v+u)$. Furthermore, $\frac14v+\frac14w=\frac12(\frac12v+\frac12w)=\frac12u$. – Daryl Jul 28 '13 at 03:03

1 Answers1

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Seems O.K., as far as the vectors go. However, since the problem says "mark the points", it may be that you're expected to draw a vector in a way that it originates at $(0,0)$. You can achieve this by translating your solution.

As for the better way, I'd say no. You obviously need $v$, and you did manage to get it quite easily.

Vedran Šego
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  • Exactly what I have in mind. I expected to draw a vector from (0,0). I can translate my solution to (0,0). Just that I feel that the other two can be easily constructed and this one seem to be more difficult than the rest. I was thinking that I was missing something. – Monkey D Luffy Jul 28 '13 at 02:57
  • Well, since you have 3 points/vectors, its probably easier to first translate $v$ and then construct what needs to be constructed. – Vedran Šego Jul 28 '13 at 03:01