Decomposing a vector into parallel and perpendicular components

[dpa]

Homework Statement

Let x,y is in Vn, such that x is not equal to zero.
Show that you can find vectors y_┴ and y_// is in Vn such that y=y_┴ + y_// and y_// is parallel to x and y_┴ is perpendicular to x.

Homework Equations

x//y => x=ky
x.y=0 if x┴y

The Attempt at a Solution

I calculated the components of y along y_┴ and y_// and showed that the synthesis of these gives y. I was adviced to use x as well. So I need different method.

By supposing y_//=kx,
y_┴=y-y_//
we write,
kx.(y-y_//)=0
and I was asked to find k and show,y=y_┴ + y_//. I have no idea how to proceed after the above line.

Thank You

 

[jbunniii]

Do you know how to project a vector onto another vector?

 

[dpa]

I was suggested not to use vector projections that are in trigonometric forms.
As for the y_||=(x.y/x^2).x form,
I know that what I can do is find
y_||=(y.kx/(kx)^2).kx [sorry, k is not bold here]
and yperp.=y-xk.,
I have no idea hence forth.

 

[jbunniii]

I was suggested not to use vector projections that are in trigonometric forms.
As for the y_||=(x.y/x^2).x form,
.

This is what I meant by vector projection. In fact, you can take k = (x.y/x^2), can't you?