If the incident ray on a surface is along the unit vector $$\widehat v\,\,,$$ the reflected ray is along the unit vector $$\widehat w\,\,$$ and the normal is along unit vector $$\widehat a\,\,$$ outwards. Express $$\widehat w\,\,$$ in terms of $$\widehat a\,\,$$ and $$\widehat v\,\,.$$

Find the equation of the plane containing the line $$2x-y+z-3=0,3x+y+z=5$$ and at a distance of $${1 \over {\sqrt 6 }}$$ from the point $$(2, 1, -1).$$

A person goes to office either by car, scooter, bus or train, the probability of which being $${1 \over 7},{3 \over 7},{2 \over 7}$$ and $${1 \over 7}$$ respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is $${2 \over 9},{1 \over 9},{4 \over 9}$$ and $${1 \over 9}$$ respectively. Given that he reached office in time, then what is the probability that he travelled by a car.

If length of tangent at any point on the curve $$y=f(x)$$ intecepted between the point and the $$x$$-axis is length $$1.$$ Find the equation of the curve.