A force F=10(a hat x)- 8 (a hat y) is applied to an object that is constrained t

Question

A force F=10(a hat x)- 8 (a hat y) is applied to an object that is constrained to travel towards increasing values of x along the path defined by y=x^2 and z=0. Find the component of F that is tangent to this path at the point (2,4,0). Attempt: I used Pythagorean's theorem to F=12.81 N since the x and y components are given. I just wasn't sure if this is what the questions is asking for, or is there something else I have do do? P.S. The (a hat x) and (a hat y) are meant to be unit vectors in the x and y directions

Explanation / Answer

slope of the tangent at 2,4,0 is tn(theta) = 4

unit vector along tangent is (hat en)= cos(theta) (hat x) +sin (theta) (hat y)

=1/sqrt(17) (hat x) +4/sqrt(17) (hat y)

so componet of force along the tamgent = F. (hat en)

=10(a hat x)- 8 (a hat y) . [1/sqrt(17) (hat x) +4/sqrt(17) (hat y)]

=10/sqrt(17) - 32/sqrt(17)= -22/sqrt(17)=-5.3357N

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