The vector projection, mathbf p, of mathbfa on mathbfb is:
mathbf p = ( mathbf a cdot mathbf(hatb) )\ mathbf (hatb) = (abs(mathbf a) abs( mathbf(hatb)) cos theta) \ mathbf (hatb)
implies mathbf p = (abs(mathbf a) cos theta) \ mathbf (hatb
Now:
mathbf a cdot mathbf b = abs(mathbf a) abs( mathbf b) cos theta
implies cos theta = (mathbf a cdot mathbf b)/( abs(mathbf a) abs( mathbf b))
implies mathbf p = abs(mathbf a) (mathbf a cdot mathbf b)/( abs(mathbf a) abs( mathbf b)) mathbf (hatb
= (mathbf a cdot mathbf b)/( abs( mathbf b)) ( mathbf b)/( abs( mathbf b))
= (mathbf a cdot mathbf b)/( abs( mathbf b)^2) mathbf b
= (langle 0,1,3 rangle cdot langle3,2,1 rangle)/( 3^2 + 2^2 + 1^2) langle3,2,1 rangle
= (5)/(14) langle3,2,1 rangle