Write the vector u as a sum of two orthogonal vectors, one of which is the vector projection of u onto v, proj, u

u = (-6,8), v = (7,1)

Take as the projection of u onto v and as the vector such that b+w =u

Step-by-step explanation:

The formula of projection  of a vector u onto a vector v is given by

, where is the dot product between vectors.

First, let b ve the projection of u onto v. Then

We want a vector w, that is orthogonal to b and that b+w = u. From this equation we have that w = u-b = (-6,8)-\frac{-17}{25}(7,1)= \frac{1}{25}(-31,217)[/tex]

By construction, we have that w+b=u. We need to check that they are orthogonal. To do so, the dot product between w and b must be zero. Recall that if we have vectors a,b  that are orthogonal then for every non-zero escalar r,k the vector ra and kb are also orthogonal. Then, we can check if w and b are orthogonal by checking if the vectors (7,1) and (-31, 217) are orthogonal.

We have that . Then this vectors are orthogonal, and thus, w and b are orthogonal.

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