How do I find two vectors that are orthogonal to each other?

In summary, to find a nonzero vector v in span {v2,v3} that is orthogonal to v3, we need to find the vector projection of v2 onto v3 and subtract it from v2. This can be found using the formula [(v2 dot v3) / Length(v3)^2] times v3. The vector v1 is not relevant to the question.
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1. Find a nonzero vector v in span {v2,v3} such that v is orthogonal to v3. Express v as a linear combination of v2 and v3


2. v1= [3 5 11] v2= [5 9 20] v3= [11 20 49]


3. I know that the dot product of v and v3 must equal zero. And that v must have components between 5 and 11, 9 and 20, and 20 and 49. But I can't find the solution. I tried using pythagoras' theorem for vectors, as well as orthogonal projections but both don't work.
 
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  • #2
What does v1 have to do with the question? Do you know how to find the vector projection of v2 on v3? If so, you can just subtract it from v2 to get the orthogonal projection vector.
 
  • #3
LCKurtz What does v1 have to do with the question? Do you know how to find the vector projection of v2 on v3? If so, you can just subtract it from v2 to get the orthogonal projection vector.

I suppose v1 doesn't have anything to do with the question.

Is the vector projection of v2 onto v3 = [(v2 dot v3) / Length(v3) squared] times v3? If so I then just subtract it from v2 and that's my answer?
 
  • #4
Thank you very much LCKurtz! It worked!
 

1. How do I determine if two vectors are orthogonal?

To determine if two vectors are orthogonal, you can use the dot product. If the dot product of two vectors is equal to 0, then they are orthogonal. Another way to check is by calculating the angle between the two vectors. If the angle is 90 degrees, then they are orthogonal.

2. Can two non-zero vectors ever be orthogonal?

No, two non-zero vectors cannot be orthogonal. For two vectors to be orthogonal, their dot product must be equal to 0. If both vectors are non-zero, their dot product will always be non-zero, meaning they are not orthogonal.

3. How do I find the orthogonal complement of a vector?

The orthogonal complement of a vector is a set of all vectors that are orthogonal to the given vector. To find the orthogonal complement of a vector, you can use the Gram-Schmidt process or the cross product. Both methods will result in a vector that is orthogonal to the given vector.

4. Can two vectors in different dimensions be orthogonal?

No, for two vectors to be orthogonal, they must be in the same dimension. This means that they must have the same number of components. For example, a 2D vector cannot be orthogonal to a 3D vector.

5. How do I find two vectors that are orthogonal to a given vector?

To find two vectors that are orthogonal to a given vector, you can use the cross product. The cross product of two vectors will result in a vector that is perpendicular to both of the original vectors. You can then use this resulting vector as one of the orthogonal vectors and find the other one using the Gram-Schmidt process.

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