Let say I have 2 vectors (1,0,0) and (0,2,0), and I want to find a...

Gretchen Schwartz

Gretchen Schwartz



Let say I have 2 vectors (1,0,0) and (0,2,0), and I want to find a third vector that is orthogonal to both of them. I can do a cross product and get (0,0,2). However, I know there are infinite vector in the following form ( 0 , 0 , x ) where x R that are orthogonal to the other two. My question is what is the difference between the orthogonal vector results from cross product and any other orthogonal vector? Why the cross product gives only 1 specific orthogonal vector? What is the significance of this vector? Thank you!

Answer & Explanation

Adolfo Rich

Adolfo Rich


2022-07-07Added 9 answers

The vector you get by performing the cross product is the unique vector orthogonal to both of your original vectors that

1. has a length equal to the magnitude of the area of the parallelogram (actually rectangle in this case) with sides (1,0,0) and (0,2,0) and
2. forms a right-handed set with (1,0,0) and (0,2,0)

If you don't care about either of those two properties, then you could just choose any vector of the form ( 0 , 0 , c ). But sometimes those properties are useful.
Joel French

Joel French


2022-07-08Added 10 answers

If a and b are two vectors then the vector a × b is another vector perpendicular, or normal, the plane formed by a and b . As you rightly suspected, there are an infinite number of vectors normal to the plane formed by a and b . They are all of the form α ( a × b ), where α is a scalar. Since all of these "point" to the same direction, it is customary to choose a vector that has unit magnitude. It is given by
c = a × b | a × b | ,
where | | denotes the magnitude of a vector.

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?