# Decompose the vector $\vec v = (-3,4,-5)$ parallel and perpendicular to a plane

I have the vecotr:

$$\vec v = (-3,4,-5)$$

And the plane: $$\pi:\\x=1-\lambda\\y=-2\\z=\lambda -\mu$$

I need to decompose the vector $\vec v$ as the sum of a vector perpendicular to the plane and the other vector parallel to it. I tried projecting $\vec v$ into a vector of the plane and also projecting it into the normal of the plane, but this is not the answer. I think that even if I Project the $\vec v$ into these two vectors, I still need a third componente such that the sum of these $3$ vectors will be $\vec v$.

What am I doing wrong?

• You have done the right thing. After you find one of these, say $v_{||}$ find $v_{\perp}=v-v_{||}$. Oct 2 '14 at 0:00
• Since the plane is defined by $y=-2$, you can use $\vec{v}=(0,4,0)+(-3,0,-5)$. Oct 2 '14 at 0:19

Instead, start by projecting $\vec v$ into a normal of the plane, such as $(-1,0,1).$ This will give you the perpendicular component $\vec v_\perp.$ Letting $\vec v_{||}=\vec v-\vec v_\perp,$ you should have that $\vec v_{||}$ is parallel to the plane, and that $\vec v=\vec v_{||}+\vec v_\perp.$