Find the vector projection of u onto v. Then write u as the sum of two orthogonal​ vectors, one of which is proj (Subscript v)u.

well, the projection of u onto v is

(u.v)/|v|

Then recall that v = |u|cosθ v/|v|
and the orthogonal component of u is |u|sinθ w/|w|
where w is orthogonal to v

To find the vector projection of u onto v, you can use the following formula:

proj_v(u) = (u · v / ||v||^2) * v

where u · v represents the dot product of u and v, and ||v||^2 represents the magnitude of v squared.

To write u as the sum of two orthogonal vectors, one of which is proj_v(u), you can use the following formula:

u = proj_v(u) + u⊥

where u⊥ represents the orthogonal component of u with respect to v.

To find u⊥, you can simply subtract proj_v(u) from u:

u⊥ = u - proj_v(u)

By substituting the formula for proj_v(u) into the equation, you can express u in terms of the vector projection and the orthogonal component:

u = (u · v / ||v||^2) * v + (u - (u · v / ||v||^2) * v)

Simplifying this equation will give you the desired expression for u as the sum of two orthogonal vectors.