Finding Orthogonal Vectors in 4 Space

• shane1
In summary, to find two vectors of norm 1 that are orthogonal to the given three vectors, one can set up a system of equations or use the determinant method in 4-space. The determinant method involves finding an n-dimensional analog of the cross product, which will give a vector perpendicular to the n-1 given vectors.
shane1
I have this question that says:
Find two vectors of norm 1 that are orthagonal to the three vectors u = (2, 1, -4, 0), v = (-1, -1, 2, 2), and w = (3, 2, 5, 4).

I've tried setting up a system of equations to solve.
2a + b - 4c = 0
-a - b + 2c + 2d = 0
3a + 2b + 4c + 4d = 0

But when I did that I was left with a free variable. So basically I was wondering if there's another way to do it such as taking the determinate like how you do in 3 space. Except in 4 space.
Eg.
i j k
0 1 0
1 2 5

Shane

There will be a whole line of vectors perpendicular to those vectors. But only 2 will have norm 1.

If you know how to calculate the determinat of an nxn matrix there is an n-dimensional analog of the cross product:
$$\vec{v}=\left| \begin{array}{c c c c} \hat{i} & \hat{j} & \hat{k} & \hat{l} \\ 2 & 1 & -4 & 0 \\ -1 & -1 & 2 & 2 \\ 3 & 2 & 5 & 4 \end{array} \right |$$

Which will give you a vector perpendicular to the n-1 you already have.

1. What are orthagonal vectors in 4 space?

Orthagonal vectors in 4 space are a set of four dimensional vectors that are perpendicular to each other. This means that the dot product of any two vectors in the set is equal to 0.

2. How are orthagonal vectors in 4 space used in science?

Orthagonal vectors in 4 space are used in various fields of science, such as physics, engineering, and computer science. They are particularly useful in solving problems involving motion, forces, and geometric transformations.

3. How do you determine if two vectors in 4 space are orthagonal?

In order to determine if two vectors in 4 space are orthagonal, you can use the dot product formula. If the dot product between the two vectors is equal to 0, then they are orthagonal. Another way to determine orthagonality is by checking if the angle between the two vectors is 90 degrees.

4. What is the significance of orthagonal vectors in 4 space?

Orthagonal vectors in 4 space have many important applications in science and technology. They can be used to represent and manipulate complex systems and multi-dimensional data. They also play a crucial role in linear algebra and are essential in understanding vector spaces and transformations.

5. Can orthagonal vectors exist in higher dimensions?

Yes, orthagonal vectors can exist in any number of dimensions. In fact, the concept of orthagonality can be extended to any number of dimensions and is not limited to just 4 space. However, it becomes increasingly difficult to visualize or work with vectors in higher dimensions, making 4 space a common choice for practical applications.

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