Question-84

If \(\displaystyle u,v,w\) are distinct vectors in \(\displaystyle \mathbb{R}^{2}\) that are pairwise orthogonal, which of the following are true?

• \(\displaystyle u,v,w\) are linearly independent
• \(\displaystyle u,v,w\) are linearly dependent
• Each of \(\displaystyle \{u,v\}\), \(\displaystyle \{v,w\}\) and \(\displaystyle \{u,w\}\) is a basis for \(\displaystyle \mathbb{R}^{2}\)
• \(\displaystyle \text{span}\{u,v,w\} =\mathbb{R}^{3}\)

Hint

Are all three vectors non-zero?

Answer

• \(\displaystyle u,v,w\) are linearly independent
• \(\displaystyle u,v,w\) are linearly dependent
• Each of \(\displaystyle \{u,v\}\), \(\displaystyle \{v,w\}\) and \(\displaystyle \{u,w\}\) is a basis for \(\displaystyle \mathbb{R}^{2}\)
• \(\displaystyle \text{span}\{u,v,w\} =\mathbb{R}^{3}\)

Solution

If \(\displaystyle u,v,w\) are distinct vectors that are pairwise orthogonal, one of them has to be zero. If none of them are zero, then \(\displaystyle \{u,v,w\}\) is independent, which can’t be the case since these three vectors are in \(\displaystyle \mathbb{R}^{2}\).

Without loss of generality, let \(\displaystyle u=0\). Then, \(\displaystyle v,w\) are non-zero. We see that \(\displaystyle \{u,v,w\}\) is linearly dependent. Only \(\displaystyle \{v,w\}\) is a basis of \(\displaystyle \mathbb{R}^{2}\) and \(\displaystyle \text{span}\{u,v,w\} =\mathbb{R}^{2}\).