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Orthogonal and Orthonormal Vectors in Linear Algebra

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  • Difficulty Level : Medium
  • Last Updated : 01 Apr, 2022

Orthogonal Vectors: Two vectors are orthogonal to each other when their dot product is 0. How do we define the dot product? Dot product(scalar product) of two n-dimensional vectors A and B, is given by this expression.A . B=\sum_{i=1}^{n} a_{i} b_{i}Thus the vectors A and B are orthogonal to each other if and only ifA.B=\sum_{i=1}^{n} a_{i} b_{i}=A^{T} B=0Note: In a compact form the above expression can be written as (A^T)B. Example: Consider the vectors v1 and v2 in 3D space.v_{1}=\left[\begin{array}{c} 1 \\ -2 \\ 4 \end{array}\right], v_{2}=\left[\begin{array}{l} 2 \\ 5 \\ 2 \end{array}\right]Taking the dot product of the vectors.v_{1}, v_{2}=V_{1}^{T} V_{2}=[1-24]\left[\begin{array}{l} 2 \\ 5 \\ 2 \end{array}\right]=0Hence the vectors are orthogonal to each other. Code: Python program to illustrate orthogonal vectors. 

Python3




# A python program to illustrate orthogonal vector
  
# Import numpy module
import numpy
  
# Taking two vectors
v1 = [[1, -2, 4]]
v2 = [[2, 5, 2]]
  
# Transpose of v1
transposeOfV1 = numpy.transpose(v1)
  
# Matrix multiplication of both vectors
result = numpy.dot(v2, transposeOfV1)
print("Result  = ", result)
  
# This code is contributed by Amiya Rout

Output:

Result = [[0]]

Unit Vector: Let’s consider a vector A. The unit vector of the vector A may be defined as\hat{a}=\frac{A}{|A|}Let’s understand this by taking an example. Consider a vector A in 2D space.A=\left[\begin{array}{l} 3 \\ 4 \end{array}\right]The magnitude of A is given by\text { Magnitude of } \mathrm{A}:|A|=\sqrt{3^{2}+4^{2}}=5So the unit vector of A can be calculated as\hat{a}=\frac{A}{|A|}=\left[\begin{array}{l} 3 / 5 \\ 4 / 5 \end{array}\right]Properties of unit vector:

  • Unit vectors are used to define directions in a coordinate system.
  • Any vectors can be written as a product of a unit vector and a scalar magnitude.

Orthonormal vectors: These are the vectors with unit magnitude. Now, take the same 2 vectors which are orthogonal to each other and you know that when I take a dot product between these 2 vectors it is going to 0. So If we also impose the condition that we want each of these vectors to have unit magnitude then what we could possibly do is by taking this vector and then divide this vector by the magnitude of this vector as we see in the unit vector. Now we can write v1 and v2 asv_{1}=\left[\begin{array}{c} 1 \\ -2 \\ 4 \end{array}\right] / \sqrt{1^{2}+(-2)^{2}+4^{2}} \quad v_{2}=\left[\begin{array}{l} 2 \\ 5 \\ 2 \end{array}\right] / \sqrt{2^{2}+5^{2}+2^{2}}So what we do is we have taken the vectors from the previous example and converted them into unit vectors by dividing them by their magnitudes. So, these vectors will still be orthogonal to each other and now individually they also have unit magnitude. Such vectors are known as orthonormal vectors. Note: All orthonormal vectors are orthogonal by the definition itself.


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