Question

If the points $\overline{a}+\overline{b},\overline{a}-\overline{b},\overline{a}+k\overline{b}$ are collinear, than

• A. $k$ has only one real value
• B. $k$ has two real value
• C. $k$ has all real values
• D. $k$ has finite number of real values

Answer 1

The correct option is C $k$ has all real values

$\overline{a}+\overline{b},\overline{a}-\overline{b}$ and $\overline{a}+k\overline{b}$ are collinear.

$(\overline{a}+\overline{b})-(\overline{a}-\overline{b})=\lambda \left(\right(\overline{a}+k\overline{b})-(\overline{a}+\overline{b}\left)\right)$

$2\overline{b}=\lambda (k-1)\overline{b}$

Depending on $\lambda ,k$ can how several solutions.

This is true goes all real value of k.