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 $$\hat{i}+\hat{j}$$ and the orthogonal would be $$\hat{i} – \hat{j}$$

using projection of F onto the parallel and orthogonal

$$\frac{<1,1>\cdot<2,1>}{||<1,1>||^2}<1,1> = <\frac{3}{2}, \frac{3}{2} >$$

$$\frac{<1,-1>\cdot<2,1>}{||<1,-1>||^2}<1,-1> = <\frac{1}{2} , \frac{-1}{2}>$$

$$\vec{F} = <\frac{3}{2}, \frac{3}{2} > + <\frac{1}{2}, \frac{-1}{2} >$$

$$= 2\hat{i} + 1\hat{j}$$

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