Let \(\ell_0\) be the line defined by the vector equation \(\hat{i}+2 \hat{j}+3 \hat{k}+\lambda(\hat{i}+\hat{j}+\hat{k})\), with \(\lambda\) real. Which of the following vector equations, with \(\mu\) real, defines a line which intersects \(\ell_0\) ?

• A

  \(2 \hat{i}+3 \hat{j}+\mu(\hat{i}-\hat{j})\).
• B

  \(3 \hat{i}-2 \hat{j}+\hat{k}+\mu(-\hat{i}+\hat{j})\).
• Correct answer
  C

  \(\hat{i}+3 \hat{j}+5 \hat{k}+\mu(2 \hat{i}+3 \hat{j}+4 \hat{k})\).
• D

  \(-\hat{i}-2 \hat{j}-3 \hat{k}+\mu(-\hat{i}-\hat{j}-\hat{k})\).

Hence, the correct answer is option: 'C'.