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I do not know how to approach this problem. Any hints will be helpful:

Let $u,\ v,\ w \;$ be three points in $\mathbb{R}^3$ not lying in any plane containing the origin.

Then which of the following are true?:

  1. $\alpha_1 u + \alpha_2 v + \alpha_3 w = 0 \implies \alpha_1 = \alpha_2 =\alpha_3 = 0$
  2. $u,\ v, \ w\;$ are mutually orthogonal
  3. one of $u, \ v, \ w\;$ has to be zero
  4. $u, \ v, \ w\;$ cannot be pairwise orthogonal
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The question is maybe to choose which of 1,2,3,4 must be true, given that u,v,w not on a plane containing the origin. Is that it? In that case it looks like choice 1. –  coffeemath Nov 19 '12 at 15:48
    
@coffeemath yes,you are right. –  learner Nov 19 '12 at 15:55
    
budha: I transcribed your question, from the image of it, directly into the post. Try, when possible, to type out your questions directly into the body of your questions; image links can "get lost" over time, and for reasons related to accessibility (for the blind, e.g.), it's best to type out questions (and formating in LaTeX is always good, too!). –  amWhy Nov 19 '12 at 17:17

1 Answer 1

When encountering problems like this, it often helps to start by eliminating those options that cannot be true and ruling out those that are not necessarily true (by finding counterexamples which show a particular choice doesn't necessarily hold):

  • $(3)$ you can see that $(3)$ cannot be true: recall that you are given the premise that $u, v, w$ do not lie on a plane containing $(0, 0, 0)$. So no linear combination of the vectors $u, v, w$ will be zero, unless each is multiplied by the scalar $0$. (Note, the last sentence is essentially what option $(1)$ is stating.)

  • $(4)$ Similarly, there is no reason why $(4)$ would necessarily be true. Can you think of a counterexample? All you need to do is think of one counterexample which satisfies the premises, but for which the option does not hold. That is, you can rule it out.

  • $(2)$ Are you given enough information to conclude that the vectors are mutually orthogonal? Think, e.g., of a counterexample. What is required for three vectors to be mutually orthogonal? Given what you're told in the problem statement, is it necessarily the case that $u, v, w$ are mutually orthogonal? If not, rule it out.

This brings us to option $(1)$:

  • $(1)$ is stating, essentially, that the vectors $u,\ v, \ \text{and}\;w\;$ are linearly independent. If you understand that your vectors are linearly independent, then $(1)$ follows.
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