# 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...
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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!

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