Dot/Scalar/Inner Product of Vectors, Magnitude of Vectors and Unit Vectors

1. Dot Product of 2 Vectors that are given in terms of Standard Basis can be calculated Only if both the Vectors have the Same Number of Components. In such cases, the Dot Product is calculated by Multiplying the Corresponding Components of 2 Vectors and Adding up All the Products. For example lets consider 2 Vectors \(A\) and \(B\) given in terms of Standard Basis each having \(N\) Components as given in the following
   
   \(A= \begin{bmatrix}a_1\\a_2\\ \vdots \\a_n \end{bmatrix}\hspace{.5cm} B= \begin{bmatrix}b_1\\b_2\\ \vdots \\b_n \end{bmatrix}\)
   
   The Dot Product of \(A\) and \(B\) is calculated as follows
   
   \(A \cdot B = B \cdot A = a_1b_1 + a_2b_2 + \cdots + a_nb_n\)
   
   If Either \(A\) Or \(B\) or Both are Complex Vectors, the Dot Product can also be calculated in 3 more ways as follows
   1. If \(A\) is a Complex Vector, the Dot Product between Vectors \(\overline{A}\) (Conjugate of Vector \(A\) ) and \(B\) is calculated as follows:
      
      \(\overline{A} \cdot B = B \cdot \overline{A} = \overline{a_1}b_1 + \overline{a_2}b_2 + \cdots + \overline{a_n}b_n\)   (where \(\overline{a_1}, \overline{a_2}, \cdots \overline{a_n}\) are Complex Conjugates of \(a_1, a_2, \cdots a_n\) respectively)
   2. If \(B\) is a Complex Vector, the Dot Product between Vectors \(A\) and \(\overline{B}\) (Conjugate of Vector \(B\) ) is calculated as follows:
      
      \(A \cdot \overline{B} = \overline{B} \cdot A = a_1\overline{b_1} + a_2\overline{b_2} + \cdots + a_n\overline{b_n}\)   (where \(\overline{b_1}, \overline{b_2}, \cdots \overline{b_n}\) are Complex Conjugates of \(b_1, b_2, \cdots b_n\) respectively)
   3. If Both \(A\) and \(B\) are a Complex Vectors, the Dot Product between Vectors \(\overline{A}\) (Conjugate of Vector \(A\) ) and \(\overline{B}\) (Conjugate of Vector \(B\) ) is calculated as follows:
      
      \(\overline{A} \cdot \overline{B} = \overline{B} \cdot \overline{A} = \overline{a_1}\overline{b_1} + \overline{a_2}\overline{b_2} + \cdots + \overline{a_n}\overline{b_n}\)
   The values of \(A \cdot B\) (or \(B \cdot A\) ) and \(\overline{A} \cdot \overline{B}\) (or \(\overline{B} \cdot \overline{A}\)) are Complex Conjugates of each other.
   
   The values of \(\overline{A} \cdot B\) (or \(B \cdot \overline{A}\) ) and \(A \cdot \overline{B}\) (or \(\overline{B} \cdot A\)) are Complex Conjugates of each other.
   
   Since the Result of Dot Product of Vectors is a Scalar Value, it is also called Scalar Product of Vectors.
2. Dot Product between 2 Vectors \(A\) and \(B\) as given above can also be represented as a Matrix Multiplication between Transpose of One Vector and the Other Vector as follows
   
   \(A \cdot B = A^TB = \begin{bmatrix}a_1&a_2& \cdots &a_n \end{bmatrix} \begin{bmatrix}b_1\\b_2\\ \vdots \\b_n \end{bmatrix} = B \cdot A = B^TA=\begin{bmatrix}b_1&b_2& \cdots &b_n \end{bmatrix} \begin{bmatrix}a_1\\a_2\\ \vdots \\a_n \end{bmatrix}= a_1b_1 + a_2b_2 + \cdots + a_nb_n\)
   
   If \(A\) is a Complex Vector, the Dot Product \(\overline{A} \cdot B\) (or \(B \cdot \overline{A}\) ) can also be represented as a Matrix Multiplication between Conjugate Transpose of Vector \(A\) and Vector \(B\) as follows
   
   \(\overline{A} \cdot B =B \cdot \overline{A}=A^\dagger B = \begin{bmatrix}\overline{a_1}&\overline{a_2}& \cdots &\overline{a_n} \end{bmatrix} \begin{bmatrix}b_1\\b_2\\ \vdots \\b_n \end{bmatrix}= \overline{a_1}b_1 + \overline{a_2}b_2 + \cdots + \overline{a_n}b_n\)
   
   If \(B\) is a Complex Vector, the Dot Product \(A \cdot \overline{B}\) (or \(\overline{B} \cdot A\) ) can also be represented as a Matrix Multiplication between Conjugate Transpose of Vector \(B\) and Vector \(A\) as follows
   
   \(A \cdot \overline{B}=\overline{B} \cdot A =B^\dagger A = \begin{bmatrix}\overline{b_1}&\overline{b_2}& \cdots &\overline{b_n} \end{bmatrix} \begin{bmatrix}a_1\\a_2\\ \vdots \\a_n \end{bmatrix} = a_1\overline{b_1} + a_2\overline{b_2} + \cdots + a_n\overline{b_n}\)
   
   Since these are same as Inner Product of Matrices, the Dot Product of Vectors is also called Inner Product of Vectors.
3. The Magnitude/Norm/Length of Any Vector \(A\) given in terms of Standard Basis is given by the Positive Real Square Root Value or Complex Square Root Values of the Dot Product of the Vector with itself as follows
   
   Magnitude/Norm/Length of Any Vector \(A = \sqrt{A \cdot A} = \sqrt{a_1a_1 + a_2a_2 + \cdots + a_na_n}\)
   
   The Magnitudes/Norms/Lengths calculated this way are Positive Real Values for Real Vectors and are Complex Values for Complex Vector.
4. The Magnitude/Norm/Length of Any Complex Vector \(A\) given in terms of Standard Basis is also given by the Positive Square Root Value of the Dot Product of the Vector with it's Conjugate as follows
   
   Magnitude/Norm/Length of Any Complex Vector \(A = \sqrt{A \cdot \overline{A}} = \sqrt{\overline{A} \cdot A} = \sqrt{\overline{a_1}a_1 + \overline{a_2}a_2 + \cdots + \overline{a_n}a_n}\)
   
   The Magnitudes/Norms/Lengths calculated this way are Always Positive Real Values.
5. Any Vector of any Magnitude can be converted into a Vector of Magnitude 1 by Dividing each of its Components with the Magnitude of the Vector. Any Vector with Magnitude 1 is called a Unit Vector.
6. Following example demonstrates the calculation of Dot Product, Magnitude of Vectors and Conversion of Vectors to Unit Vectors for 2 Real Vectors \(A\) and \(B\) given in terms of Standard Basis
   
   \(A= \begin{bmatrix}-3\\2\\7\\-5\end{bmatrix}\hspace{.5cm} B= \begin{bmatrix}5\\-1\\4\\8\end{bmatrix}\)
   
   The Dot Product of \(A\) and \(B\) are calculated as follows
   
   \(A \cdot B = B \cdot A = -3 \cdot 5 + 2 \cdot -1 + 7 \cdot 4 + -5 \cdot 8 = -29\)
   
   The Magnitude of Vectors \(A\) and \(B\) are calculated as follows
   
   Magnitude of \(A = \sqrt{A \cdot A} = \sqrt{-3 \cdot -3 + 2 \cdot 2 + 7 \cdot 7 + -5 \cdot -5 }=\sqrt{87}\)
   
   Magnitude of \(B = \sqrt{B \cdot B} = \sqrt{5 \cdot 5 + -1 \cdot -1 + 4 \cdot 4 + 8 \cdot 8}=\sqrt{106}\)
   
   The Unit Vectors Corresponding to Vectors \(A\) and \(B\) are given by Dividing their Components by their Respective Magnitudes as given in the following
   
   Unit Vector Corresponding to Vector \(A= \begin{bmatrix}\frac{-3}{\sqrt{87}}\\\frac{2}{\sqrt{87}}\\\frac{7}{\sqrt{87}}\\\frac{-5}{\sqrt{87}}\end{bmatrix}\)
   
   Unit Vector Corresponding to Vector \(B= \begin{bmatrix}\frac{5}{\sqrt{106}}\\\frac{-1}{\sqrt{106}}\\\frac{4}{\sqrt{106}}\\\frac{8}{\sqrt{106}}\end{bmatrix}\)
7. Following example demonstrates the calculation of Dot Product, Magnitude of Vectors and Conversion of Vectors to Unit Vectors for 2 Complex Vectors \(C\) and \(D\) given in terms of Standard Basis
   
   \(C= \begin{bmatrix}2-5i\\1+7i\\3+i\end{bmatrix}\hspace{.5cm} D= \begin{bmatrix}-3-5i\\2+i\\4-6i\end{bmatrix}\)
   
   The Dot Products between \(C\) and \(D\) are calculated as follows
   
   \(C \cdot D = D \cdot C= (2-5i) \cdot (-3-5i) + (1+7i) \cdot (2+i) + (3+i) \cdot (4-6i) = -18+6i\)
   
   \(C \cdot \overline{D} = \overline{D} \cdot C = (2-5i) \cdot (-3+5i) + (1+7i) \cdot (2-i) + (3+i) \cdot (4+6i) = 34+60i\)
   
   \(\overline{C} \cdot D = D \cdot \overline{C} = (2+5i) \cdot (-3-5i) + (1-7i) \cdot (2+i) + (3-i) \cdot (4-6i) = 34-60i\)
   
   \(\overline{C} \cdot \overline{D} = \overline{D} \cdot \overline{C} = (2+5i) \cdot (-3+5i) + (1-7i) \cdot (2-i) + (3-i) \cdot (4+6i) = -18-6i\)
   
   The Real Magnitude of Vectors \(C\) and \(D\) are calculated as follows
   
   Real Magnitude of \(C = \sqrt{C \cdot \overline{C}} = \sqrt{\overline{C} \cdot C} = \sqrt{(2+5i) \cdot (2-5i) + (1-7i) \cdot (1+7i) + (3-i) \cdot (3+i) }=\sqrt{89}\)
   
   Real Magnitude of \(D = \sqrt{D \cdot \overline{D}} = \sqrt{\overline{D} \cdot D} = \sqrt{(-3+5i) \cdot (-3-5i) + (2-i) \cdot (2+i) + (4+6i) \cdot (4-6i)}=\sqrt{91}\)
   
   The Unit Vectors Corresponding to Vectors \(C\) and \(D\) calculated by Dividing their Components by their Respective Real Magnitudes are given in the following
   
   Unit Vector Corresponding to Vector \(C\) using Real Magnitude \(= \begin{bmatrix}\frac{2-5i}{\sqrt{89}}\\\frac{1+7i}{\sqrt{89}}\\\frac{3+i}{\sqrt{89}}\end{bmatrix}\)
   
   Unit Vector Corresponding to Vector \(D\) using Real Magnitude \(= \begin{bmatrix}\frac{-3-5i}{\sqrt{91}}\\\frac{2+i}{\sqrt{91}}\\\frac{4-6i}{\sqrt{91}}\end{bmatrix}\)
   
   The Complex Magnitude of Vectors \(C\) and \(D\) are calculated as follows
   
   Complex Magnitude of \(C = \sqrt{C \cdot C} = \sqrt{(2-5i) \cdot (2-5i) + (1+7i) \cdot (1+7i) + (3+i) \cdot (3+i) }=\sqrt{-61}\)
   
   Complex Magnitude of \(D = \sqrt{D \cdot D} = \sqrt{(-3-5i) \cdot (-3-5i) + (2+i) \cdot (2+i) + (4-6i) \cdot (4-6i)}=\sqrt{-33-14i}\)
   
   The Unit Vectors Corresponding to Vectors \(C\) and \(D\) calculated by Dividing their Components by their Respective Complex Magnitudes are given in the following
   
   Unit Vector Corresponding to Vector \(C\) using Complex Magnitude \(= \begin{bmatrix}\frac{2-5i}{\sqrt{-61}}\\\frac{1+7i}{\sqrt{-61}}\\\frac{3+i}{\sqrt{-61}}\end{bmatrix}\)
   
   Unit Vector Corresponding to Vector \(D\) using Complex Magnitude \(= \begin{bmatrix}\frac{-3-5i}{\sqrt{-33-14i}}\\\frac{2+i}{\sqrt{-33-14i}}\\\frac{4-6i}{\sqrt{-33-14i}}\end{bmatrix}\)
8. Any 2 Real Vectors \(A\) and \(B\) are called Orthogonal if the Dot Product \(A \cdot B = 0\).
9. Any 2 Complex Vectors \(A\) and \(B\) are called Orthogonal if the Dot Products \(A \cdot B = 0\) or \(\overline{A} \cdot B = 0\) (which is same as \(A \cdot \overline{B} = 0\) as they are Conjugates).
10. Any 2 Vectors are called Orthonormal if they are Orthogonal and Both Vectors are Unit Vectors.