given v= 6i-8j and w=i+j. Find a. Projw v b. decompose v into 2 vectors v1 and v2, where v2 is parallel to w and v2 is orthogonal to w

Answer 1

a) proj_wv is the projection of v onto w. Use the following equation:

proj_wv = [(v•w)/((magnitude(w))²)] w

v•w = (6*1) + (-8*1) = -2

(magnitude w)² = (sqrt(1² + 1²))² = 2

proj_wv = ((-2)/2)(i + j) = -i - j

b) The two components of the decomposed v will add to create the original vector v. v₁ that is parallel to w will be the same as the projection of v onto w.

v = v₁ + v₂

v₂ = v - v₁ = (6i - 8j) - (-i -j) = 7i -7j

You can check that v₂ is orthagonal by taking the dot product (v₂•w). This equals 0.