Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Suppose we have a vector $a \in \mathcal{R}^2$ and $x,y \in \mathcal{R}^2$ and let $\alpha$ is the angle between $x$ and $y$. Define: $$z = (a \cdot x) x + (a \cdot y)y$$

If $\alpha = \pi /2$ then $z = x$, if $\alpha < \pi /2$ $z$ is longer than $x$ otherwise it is shorter.

How is the length of $z$ related with $\alpha$? Does it depends also on the relative angle between $x$ and $a$ or $y$ and $a$?

share|cite|improve this question
What does $ax$ mean? – Phira Oct 30 '12 at 17:22

If we talk about the length of z, then it is determined by

$$|z|^2 = |x|^2 + |y|^2 + 2 |x||y| \cos\alpha $$

which clearly shows that it is dependent on the length of vectors $x, y$ and angle between the two vectors $\alpha$.

For $0<\alpha<\pi/2$, $z$ is greater than $x$.

For $\pi/2<\alpha<\pi$, $z$ is less than $x$.

For $\alpha = \pi/2$, $z$ is greater than $x$ unless the length of vector $y$ is zero.

Since the product is not defined by you in the case of $ax$, then one can make assumptions about the definition of product $ax$ and $ay$. Please clarify about it.

share|cite|improve this answer
You forgot the square on $|z|^2$. – xavierm02 Oct 30 '12 at 17:37
Thanks for pointing it out. It is now corrected. – stackoverflowery Oct 30 '12 at 17:46

Your Answer


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.