# Thread: Vector projection and perpendicular

1. ## Vector projection and perpendicular

Hi,

I am trying to solve the following problem:

1. Find the projection of $\displaystyle u=2i+j+3k$ on $\displaystyle v=-i+3j+2k$. Hence resolve $\displaystyle u$ into two vectors, one parallel to $\displaystyle v$ and the other perpendicular to $\displaystyle v$.

I can solve the the first part, just by using the projection formula = $\displaystyle (u . v / |v|) . v / |v|$

But how do I get it perpendicular to $\displaystyle v$? I know that if the dot product of two vectors = 0, then they are perpendicular. So:

projection dot ??? = 0. How do I calculate what to dot it with?

2. ## Re: Vector projection and perpendicular

By trial and error i+j-k is perpendicular to v

4. ## Re: Vector projection and perpendicular

Originally Posted by M.R
1. Find the projection of $\displaystyle u=2i+j+3k$ on $\displaystyle v=-i+3j+2k$.
Hence resolve $\displaystyle u$ into two vectors, one parallel to $\displaystyle v$ and the other perpendicular to $\displaystyle v$.
I can solve the the first part, just by using the projection formula = $\displaystyle (u . v / |v|) . v / |v|$
But how do I get it perpendicular to $\displaystyle v$?
You need to know these two:

$\displaystyle {u_{||}} = \frac{{u \cdot v}}{{v \cdot v}}v\;\& \;{u_ \bot } = u - {u_{||}}$

Those two are perpendicular and their sum is $\displaystyle u$.

5. ## Re: Vector projection and perpendicular

Originally Posted by Plato
$\displaystyle \;{u_ \bot } = u - {u_{||}}$
That's what I needed. Thanks

6. ## Re: Vector projection and perpendicular

Originally Posted by M.R
Hi,

I am trying to solve the following problem:

1. Find the projection of $\displaystyle u=2i+j+3k$ on $\displaystyle v=-i+3j+2k$. Hence resolve $\displaystyle u$ into two vectors, one parallel to $\displaystyle v$ and the other perpendicular to $\displaystyle v$.

I can solve the the first part, just by using the projection formula = $\displaystyle (u . v / |v|) . v / |v|$

But how do I get it perpendicular to $\displaystyle v$? I know that if the dot product of two vectors = 0, then they are perpendicular. So:

projection dot ??? = 0. How do I calculate what to dot it with?
This is a more geometrical explanation (if I understand your problem correctly!):

The vectors $\displaystyle \vec u$ und $\displaystyle \vec v$ span a plane. You are looking for a vector $\displaystyle \vec p$ which lies in this plane and is perpendicular to $\displaystyle \vec v$.

$\displaystyle \displaystyle{\vec p = (\vec u \times \vec v) \times \vec v}$

will do.

7. ## Re: Vector projection and perpendicular

Originally Posted by earboth
This is a more geometrical explanation (if I understand your problem correctly!):

The vectors $\displaystyle \vec u$ und $\displaystyle \vec v$ span a plane. You are looking for a vector $\displaystyle \vec p$ which lies in this plane and is perpendicular to $\displaystyle \vec v$.

$\displaystyle \displaystyle{\vec p = (\vec u \times \vec v) \times \vec v}$

will do.
Would that also mean that it is perpendicular to both vectors $\displaystyle \vec u$ and $\displaystyle \vec v$?

8. ## Re: Vector projection and perpendicular

Originally Posted by M.R
Would that also mean that it is perpendicular to both vectors $\displaystyle \vec u$ and $\displaystyle \vec v$?
Yes that is correct. The answer I gave you is a standard in as much as it is the decomposition of $\displaystyle \vec u$ into the sum of two vectors one parallel to $\displaystyle \vec v$ the other perpendicular to $\displaystyle \vec v$.

Thanks