profusely hard question

Question

Let vector v = (5, 3). Give an example of a vector x other than vector v for which proj_{vector v}(vector x) = vector v

Answer 1

proj_vx = [(v•x)/llvll²] v

 

v = < 5, 3>   x = <a, b>

 

v•x = 5a + 3b     llvll² = [√(5² + 3²)]² = 34

 

We are looking for numbers a and b so that [(v•x)/llvll²]v = v

 

So, [(5a + 3b)/34](<5, 3>) = <5, 3>

 

(5a + 3b)/34 = 1

 

5a + 3b = 34  For example, let a = 2 and b = 8

 

x = <2,8> is a vector such that the projection of x onto <5,3> is <5,3>.