# equation of a 3rd vector that is coplanar to 2 other vectors that are at right angles

• January 12th 2011, 09:08 PM
jsmithson
equation of a 3rd vector that is coplanar to 2 other vectors that are at right angles
Hi folks!

I start with 2 vectors (in 3 dimensions) that are at right angles to each other (and are therefore coplanar). I want to find a third vector that lies in the same plane as the other two but at a specific angle to one of the vectors ( eg. 23 degrees to vector 2 ).

Another way to think of the problem is that I want to change the angle of one of the two vectors to something other than 90 degrees but to ensure that it remains coplanar with the original two vectors.

So to clarify - I have 3 knowns: 2 vectors and 1 angle; and I have 1 unknown: 1 vector in the same plane as the 2 other vectors at an angle of X to one of the other vectors. I'm looking for a solution to calculate the unknown.

My first thoughts were that the new vector is a line on the surface of a right cone formed with it's axis as one of the first vectors; and that one of the solutions to the intersection of the cone and the plane formed by the first two vectors will be the answer. I'm not sure how to do this mathematically - and perhaps there's a better solution.

Thanks!
• January 12th 2011, 09:22 PM
snowtea
I assume you are in 3 space.
So your original two vectors are v and w.

Find the vector n normal (perpendicular) to v and w. You can do this using the cross product.
Now, any vector normal to n is a possible vector you are looking for.
• January 13th 2011, 12:27 AM
earboth
Quote:

Originally Posted by jsmithson
Hi folks!

I start with 2 vectors (in 3 dimensions) that are at right angles to each other (and are therefore coplanar). I want to find a third vector that lies in the same plane as the other two but at a specific angle to one of the vectors ( ie. 45 degrees to vector 2 ).

1. Let $\vec u$ and $\vec v$ denote the given vectors and $\vec w$ the vector whose components you want to calculate.
Calculate the unit vectors of $\vec u$ and $\vec v$.

2. $\vec w = \dfrac{\vec u}{|\vec u|} + \dfrac{\vec v}{|\vec v|}$
$\vec w$ and $\vec v$ (or $\vec u$) include an angle of 45°

Quote:

Another way to think of the problem is that I want to change the angle of one of the two vectors to something other than 90 degrees but to ensure that it remains coplanar with the original two vectors.
Sorry, but I don't understand .... (Thinking)

Quote:

My first thoughts were that the new vector is a line on the surface of a right cone formed with it's axis as one of the first vectors; and that one of the solutions to the intersection of the cone and the plane formed by the first two vectors will be the answer. I'm not sure how to do this mathematically - and perhaps there's a better solution.

Thanks!
• January 13th 2011, 12:46 AM
jsmithson
Thanks for your input earboth. I chose poorly with my 45 degrees example - your solution is correct but only for 45 degrees. The angle will not always be 45 degrees.
• January 13th 2011, 12:49 AM
jsmithson
Thanks for your input snowtea. Unfortunately my question was not clear enough - I have modified it.
• January 13th 2011, 10:18 AM
HallsofIvy
Any three-vector that is in the same plane as two other indendent vectors (and two vectors perpendicular to one another are indpendent) can be written as a linear combination of the two vectors. That is, any vector, w, that is in the same plane as u and v can be written w= Au+ Bv. Use whatever other conditions you have on w to find the numbers A and B.
• January 14th 2011, 07:31 PM
jsmithson
Thanks for your answer hallsofivy! I can use sin and cos to work out A and B!