Pls help vectors

Couple of ways to approach it, but using the dot product = 0 is probably the easiest.
So D lies on the line and you want to determine the value of the line scalar multiplier which gives the dot product with the line direction as zero.

why do we use the product rule?

Just imagine a line which does not pass through the origin.
All points a constant distance from the origin lie on a circle (distance r from the origin).
The closest point of a line is therefore when the circle expands just enough so that the line is a tangent to a point on the circle.
That point represents the closest point D and OD is perpendicular to the line direction (zero dot product).

I'm sorry I didn't quite get that