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mds9668
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How do you find all vectors perpendicular to a set of three vectors in R^4? I know that the dot product of a normal vector and each given vector will be equal to 0. How could I set up the system of equations in matrix form?
||spoon|| said:could you not cross product??
danago said:I believe the cross product is defined only in R^3, well as far as i know :P
Defennder said:It's defined in R3 and R7 only, if I remember correctly. I don't know why, though. See here:
http://en.wikipedia.org/wiki/Cross_product#Higher_dimensions
If [tex]v_1, ..., v_{n-1} \in R^n[/tex] and [tex]\phi[/tex] is defined by
[tex]
\phi(w) = det \left( \begin{matrix}
v_1 \\
... \\
v_{n-1} \\
w
\end{matrix} \right)
[/tex]
then [tex]\phi \in \Lambda^1 (R^n) [/tex]; therefore there is a unique [tex]z \in R^n[/tex] such that
[tex]
\langle w,z \rangle = \phi(w) = det \left( \begin{matrix}
v_1 \\
... \\
v_{n-1} \\
w
\end{matrix} \right)
[/tex]
This z is denoted
[tex]
v_1 \times ... \times v_{n-1}
[/tex]
and is called the cross product of [tex]v_1, ... v_{n-1}[/tex].
mds9668 said:How do you find all vectors perpendicular to a set of three vectors in R^4? I know that the dot product of a normal vector and each given vector will be equal to 0. How could I set up the system of equations in matrix form?
A vector is considered normal to a given set of vectors if it is perpendicular to all of the vectors in that set. In other words, the dot product of the normal vector and each vector in the set is equal to zero.
To find all vectors normal to a given set of vectors, 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. By taking the cross product of each vector in the set with all the other vectors, you can find all the possible normal vectors.
Yes, there can be multiple vectors that are normal to a given set of vectors. This is because there are an infinite number of vectors that can be perpendicular to a given set of vectors. The cross product method mentioned earlier will help you find all possible normal vectors.
Finding all vectors normal to a given set of vectors is important in many areas of mathematics and science, such as physics and engineering. These normal vectors can represent forces or directions of motion in a system, and can help in solving problems involving these systems.
One limitation when finding all vectors normal to a given set of vectors is that the vectors in the set must be linearly independent. This means that none of the vectors in the set can be a scalar multiple of another vector in the set. Additionally, if the set of vectors is in three-dimensional space, then the normal vectors will also be in three-dimensional space.