SOLUTION: Calculate proj v to u. u=<2,9> and v= (-3,4) Resolve u into u1 and u2, where u1 is parallel to v and u2 is orthogonal to v.

Algebra ->  Trigonometry-basics -> SOLUTION: Calculate proj v to u. u=<2,9> and v= (-3,4) Resolve u into u1 and u2, where u1 is parallel to v and u2 is orthogonal to v.      Log On


   

Question 1042853: Calculate proj v to u. u=<2,9> and v= (-3,4)
Resolve u into u1 and u2, where u1 is parallel to v and u2 is orthogonal to v.
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
The projection of v onto u is given by
.
-------------------------------------------------------------------------
First find vector u%5B1%5D, which is the component of vector u along vector v.
This is given by %28%28u%2Av%29%2Fabs%28v%29%5E2%29v+=+%28%28-6%2B36%29%2F%289%2B16%29%29v+=+%286%2F5%29v. This is u%5B1%5D.
Now let's find u%5B2%5D. A vector orthogonal (perpendicular) to v = <-3,4> is v%5Bo%5D+ = <4,3>, since their dot product is -3*4 + 4*3 = 0.
Next find the component of u along v%5Bo%5D+.
It is given by
.
This is u%5B2%5D.
Since it is easy to verify that u%5B1%5D%2Bu%5B2%5D+=+%286%2F5%29v+%2B+%287%2F5%29v%5Bo%5D+=+u, we have found the resolution of u into its orthogonal components.