If the lines $${{x - 1} \over 2} = {{y + 1} \over 3} = {{z - 1} \over 4}$$ and $$\,{{x - 3} \over 1} = {{y - k} \over 2} = {z \over 1}$$ intersect, then the value of $$k$$ is

The value of $$k$$ such that $${{x - 4} \over 1} = {{y - 2} \over 1} = {{z - k} \over 2}$$ lies in the plane $$2x -4y +z = 7,$$ is

Let $$\alpha ,\beta ,\gamma $$ be distinct real numbers. The points with position
vectors $$\alpha \widehat i + \beta \widehat j + \gamma \widehat k,\,\,\beta \widehat i + \gamma \widehat j + \alpha \widehat k,\,\,\gamma \widehat i + \alpha \widehat j + \beta \widehat k$$