write F as a sum of an orthogonal and parallel vector

an object is moving in the direction i + j is being acted upon by the force vector 2i + j, express this force as the sum of a force in the direction of motion and a force perpendicular to the direction of motion.

the parallel would be [tex] \hat{i}+\hat{j}[/tex] and the orthogonal would be [tex]\hat{i} – \hat{j}[/tex]

using projection of F onto the parallel and orthogonal

[tex] \frac{<1,1>\cdot<2,1>}{||<1,1>||^2}<1,1> = <\frac{3}{2}, \frac{3}{2} >[/tex]

[tex] \frac{<1,-1>\cdot<2,1>}{||<1,-1>||^2}<1,-1> = <\frac{1}{2} , \frac{-1}{2}>[/tex]

[tex] \vec{F} = <\frac{3}{2}, \frac{3}{2} > + <\frac{1}{2}, \frac{-1}{2} > [/tex]

[tex] = 2\hat{i} + 1\hat{j} [/tex]

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