Consider a plane with the outer normal unit vector of (1/?2, -1?2, 0). There is

Question

Consider a plane with the outer normal unit vector of (1/?2, -1?2, 0). There is a force of (2,1,5) acting on the surface. What is the component of force normal and parallel to the plane? What is the maximum parallel component to the plane?

I would like to check my answers for the first two, but do not know how to find the maximum parallel component. I'm guessing since my direction is at a 45 degree angel, the maximum parallel component of the force should be given at the 45 degree angle?

2. Relevant equations
Dot product and Pythagorean theorem
Force = 2^2 +1^2 + 5^2 = ?30

3. The attempt at a solution
I found the normal force to be (1/2, -1/2, 0) through doing the dot product of force and the unit vector, and then multiplying the scalar product by the unit vector again.
For the parallel component, I found the magnitude of the Force and the magnitude of the normal Force, and used Pythagorean theorem. I got ?Force^2 - 1^2. = 29.98(1/?2, 1/?2, 0).

Explanation / Answer

you are solving in right way
but for maximum parallel component to the plane just use angle of 0 degree as for two parallel vector angle will 0 degree
all other solve in same way

you atempt is fine

3. The attempt at a solution
I found the normal force to be (1/2, -1/2, 0) through doing the dot product of force and the unit vector, and then multiplying the scalar product by the unit vector again.
For the parallel component, I found the magnitude of the Force and the magnitude of the normal Force, and used Pythagorean theorem. I got Force^2 - 1^2. = 29.98(1/2, 1/2, 0).

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