Orthogonal vectors, perpendicular vectors, dot product, and vector space are fundamental concepts in linear algebra. Finding the orthogonal vector to a given vector is essential for various applications, including solving systems of linear equations, finding projections of vectors onto subspaces, and performing geometric transformations.
Formula to Find Orthogonal Vector
Finding orthogonal vectors can be an important task in various mathematical applications, and understanding the process can be beneficial. Here’s a detailed walkthrough of the formula used to calculate the orthogonal vector of a given vector:
Formula for Orthogonal Vector
Let’s assume we have a vector ( \textbf{v} = [v_1, v_2, \dots, v_n] ). To find a vector ( \textbf{u} ) that is orthogonal to ( \textbf{v} ), we can use the following formula:
( \textbf{u} = \textbf{v} \times \textbf{k} )
Where:
- ( \textbf{k} ) is a unit vector in the direction of the z-axis, i.e., ( \textbf{k} = [0, 0, 1] )
Steps to Use the Formula
Here are the steps to use the formula to find the orthogonal vector:
-
Multiply each component of ( \textbf{v} ) by the corresponding component of ( \textbf{k} ).
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Calculate the cross product of the resulting vectors.
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The result is the orthogonal vector ( \textbf{u} ).
Example
Let’s find the orthogonal vector of ( \textbf{v} = [1, 2, 3] ):
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( \textbf{v} \times \textbf{k} = [1, 2, 3] \times [0, 0, 1] )
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( = [2 – 0, 0 – 1, 0 – 0] )
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( = [2, -1, 0] )
Therefore, the orthogonal vector of ( \textbf{v} ) is ( \textbf{u} = [2, -1, 0] ).
Properties of Orthogonal Vectors
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The orthogonal vector ( \textbf{u} ) is perpendicular to the given vector ( \textbf{v} ), i.e., ( \textbf{u} \cdot \textbf{v} = 0 ).
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The orthogonal vector ( \textbf{u} ) lies in the plane that is perpendicular to the given vector ( \textbf{v} ).
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The magnitude of the orthogonal vector is equal to the magnitude of the given vector, i.e., ( \lVert \textbf{u} \rVert = \lVert \textbf{v} \rVert ).
Table of Orthogonal Vectors
Here’s a table summarizing the orthogonal vectors for some commonly used vectors:
| Given Vector | Orthogonal Vector |
|---|---|
| ( \textbf{i} = [1, 0, 0] ) | ( \textbf{k} = [0, 0, 1] ) |
| ( \textbf{j} = [0, 1, 0] ) | ( -\textbf{i} = [1, 0, 0] ) |
| ( \textbf{k} = [0, 0, 1] ) | ( \textbf{j} = [0, 1, 0] ) |
| ( [1, 1, 0] ) | ( [0, 0, 1] ) |
| ( [0, -1, 1] ) | ( [2, 0, 1] ) |
Question 1:
How can we find the orthogonal vector of a given vector?
Answer:
The orthogonal vector of a given vector v is a vector w that is perpendicular to v. To find the orthogonal vector, we can use the cross product operation in three dimensions or the dot product operation in higher dimensions.
Question 2:
What are the applications of finding the orthogonal vector?
Answer:
Finding orthogonal vectors has numerous applications in geometry, physics, and engineering, including determining the direction of a force perpendicular to a surface, finding the normal to a plane or curve, and calculating the cross-sectional area of a parallelogram or parallelepiped.
Question 3:
How does the Gram-Schmidt orthogonalization process help in finding the orthogonal vector?
Answer:
The Gram-Schmidt orthogonalization process provides a systematic way of finding orthogonal vectors from a set of given vectors. It iteratively subtracts the projections of the current vector onto the previous vectors, resulting in a new vector that is orthogonal to the span of the previous vectors.
Thanks for reading! I hope this article has helped you to understand how to find the orthogonal vector. If you have any further questions, please don’t hesitate to leave a comment below. And be sure to check back later for more math-related articles.