Velocity After Ricochet -- Vector Projection (calculus 3 problem)?

The top section of text in the image is the problem, and the bottom section is the solution. I was able to get up to the point where it says the velocity after the ricochet is w = v - 2proj_(n)v. I know that v - proj_(n)v is the portion of v that is perpendicular to the normal vector, but I'm not sure why... show more The top section of text in the image is the problem, and the bottom section is the solution. I was able to get up to the point where it says the velocity after the ricochet is w = v - 2proj_(n)v. I know that v - proj_(n)v is the portion of v that is perpendicular to the normal vector, but I'm not sure why multiplying the projection by 2 will give the velocity. Any help would be very appreciated!