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  1. There is an infinity of orthogonal matrices $Q=[q_1,q_2,q_3]$ that have as the first two columns the vectors $q_1=\frac{1}{\sqrt{6}}(-1,2,-1)^T,q_2=\frac{1}{\sqrt{3}}(1,1,1)^T$.
    I would say this is not true since there is only one third vector orthogonal on other two.
  2. If $\langle u,v\rangle \ge 0$ then the measure of the angle between $u$ and $v$ is less than $\frac{\pi}{2}$
    To me it looks true, since the measure can take values only in interval (0,1)
  3. The projection of a line on a plane is always a line.
    True?

Thank you for taking your time.

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    @ArthurFischer For (1), Orthogonal means Orthonormal columns...So no scalar multiples (excepting by -1) are allowed....2012-01-18

1 Answers 1

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Ok, let's see:

  1. I agree with you, false. But there is not only one. Keep in mind that the determinant has to be $\pm1$ and not only $1$.

  2. What happens when $\langle u,v\rangle$ is exactly $0$? Keep in mind that $\langle u,v\rangle=\|u\|\|v\|\cos\angle(u,v)$. And the measure is not in the interval $(0,1)$, since it's an angle and not the cosine.

  3. What if the line is orthogonal to the plane.

I guess the main intention of these questions is to make you read them carefully and precisely and not judge by just pure intuition. Since they prove false when looking at the special cases after seeming obviously true at first.