Cross Product of Vectors

1. Cross Product of Vectors is carried out Between 2 Vectors of 3 Dimensions each.
2. Cross Product of 2 Vectors is also known as Vector Products of Vectors as it generates another Vector. The Resultant Vector of a Cross Product of 2 Vectors is Perpendicular to both the Participating Vectors. The Direction of the Resultant Vector is given by the Right Hand Thumb Rule, which in case of Cross Products states that if we Curl the Fingers of our Right Hand while Sticking the Thumb out, the curling of the fingers denotes the Counter-Clockwise Rotation of the 1st Vector towards the 2nd Vector by Angle \(\theta\) (such that \(0\leq \theta \leq 180^\circ\)) and the Thumb gives the Direction of the Resultant Vector
3. Cross Product of a Vector with itself or between any 2 Parallel Vectors is a NULL Vector.
4. Cross Product of 2 Vectors is Anti-Commutative. That is, for any 2 Non-Parallel Vectors \(\vec{A}\) and \(\vec{B}\)
   
   \(\vec{A} \times \vec{B} = - \vec{B} \times \vec{A}\)
5. In 3-Dimensional Cartesian Coordinate System any Vector is by default Directionally Represented on the Basis of the Standard Basis Vectors given by Unit Vectors \(\mathbf{\hat{i}}\), \(\mathbf{\hat{j}}\) and \(\mathbf{\hat{k}}\). Because of Perpendicular Result Vector property of Cross Products
   
   \(\hat{i} \times \hat{j} = \hat{k},\hspace{.5cm} \hat{j} \times \hat{k} = \hat{i},\hspace{.5cm} \hat{k} \times \hat{i} = \hat{j}\)
   
   Also, because of Anti Commutative property of Cross Product
   
   \(\hat{j} \times \hat{i} = -\hat{k},\hspace{.5cm} \hat{k} \times \hat{j} = -\hat{i},\hspace{.5cm} \hat{i} \times \hat{k} = -\hat{j}\)
   
   And, since Cross Product of a Vector with itself is Null Vector, therefore
   
   \(\hat{i} \times \hat{i} = 0,\hspace{.5cm} \hat{j} \times \hat{j} = 0,\hspace{.5cm} \hat{k} \times \hat{k} = 0\)
6. The following demonstrates the calculation of Cross Product for 2 3-Dimensional Vectors \(\vec{A}\) and \(\vec{B}\) Represented in Identity Orthonormal Basis in Cartesian Coordinate System
   
   \(\vec{A}=A_1\hat{i} + A_2\hat{j} + A_3\hat{k}\hspace{.5cm}\vec{B}=B_1\hat{i} + B_2\hat{j} + B_3\hat{k}\)
   
   \(\vec{A} \times \vec{B} =(A_1\hat{i} + A_2\hat{j} + A_3\hat{k}) \times (B_1\hat{i} + B_2\hat{j} + B_3\hat{k})\)
   
   \(\Rightarrow \vec{A} \times \vec{B} =A_1B_1(\hat{i} \times \hat{i}) + A_1B_2(\hat{i} \times \hat{j}) + A_1B_3(\hat{i} \times \hat{k}) + A_2B_1(\hat{j} \times \hat{i}) + A_2B_2(\hat{j} \times \hat{j}) + A_2B_3(\hat{j} \times \hat{k}) + A_3B_1(\hat{k} \times \hat{i}) + A_3B_2(\hat{k} \times \hat{j}) + A_3B_3(\hat{k} \times \hat{k}) \)
   
   Replacing the result of Cross Product of Unit Vectors \(\mathbf{\hat{i}}\), \(\mathbf{\hat{j}}\) and \(\mathbf{\hat{k}}\) in above equation we get
   
   \(\vec{A} \times \vec{B} =A_1B_2\hat{k} - A_1B_3\hat{j} - A_2B_1\hat{k} + A_2B_3\hat{i} + A_3B_1\hat{j} - A_3B_2\hat{i}\)
   
   \(\Rightarrow \vec{A} \times \vec{B} = (A_2B_3- A_3B_2)\hat{i} + (A_3B_1- A_1B_3)\hat{j} + (A_1B_2-A_2B_1)\hat{k}\)
   
   The calculation of Cross Product given above can be given in form of Determinant as the following
   
   \(\vec{A} \times \vec{B} =\begin{vmatrix}\mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}}\\A_1 & A_2 & A_3\\B_1 & B_2 & B_3\end{vmatrix}\)
   
   
7. The Cross Product \(\vec{A} \times \vec{B}\) between Vectors \(\vec{A}\) and \(\vec{B}\) (as defined above) can also be calculated as a Matrix Product between Skew-Symmetric Matrix Corresponding to Vector \(\vec{A}\) and Vector \(\vec{B}\). Similarly, the Cross Product \(\vec{B} \times \vec{A}\) between Vectors \(\vec{A}\) and \(\vec{B}\) can also be calculated as a Matrix Product between Skew-Symmetric Matrix Corresponding to Vector \(\vec{B}\) and Vector \(\vec{A}\). These are demonstrated below
   
   \(\vec{A} \times \vec{B} =[\vec{A}_{\mathbf{\times}}]\vec{B}\hspace{.8cm}\vec{B} \times \vec{A} =[\vec{B}_{\mathbf{\times}}]\vec{A}\)
   
   In the above equations \(\vec{A}_{\mathbf{\times}}\) and \(\vec{B}_{\mathbf{\times}}\) are Skew-Symmetric Matrices corresponding to Vectors \(\vec{A}\) and \(\vec{B}\) respectively and are given as
   
   \(\vec{A}_{\mathbf{\times}}=\begin{bmatrix}0 & -A_3 & A_2\\A_3 & 0 & -A_1\\-A_2 & A_1 & 0\end{bmatrix}\hspace{.5cm}\vec{B}_{\mathbf{\times}}=\begin{bmatrix}0 & -B_3 & B_2\\B_3 & 0 & -B_1\\-B_2 & B_1 & 0\end{bmatrix}\)
   
   Hence, the Matrix Product \([\vec{A}_{\mathbf{\times}}]\vec{B}\) for Cross Product \(\vec{A} \times \vec{B}\) is calculated as
   
   \(\vec{A} \times \vec{B}=[\vec{A}_{\mathbf{\times}}]\vec{B}=\begin{bmatrix}0 & -A_3 & A_2\\A_3 & 0 & -A_1\\-A_2 & A_1 & 0\end{bmatrix}\begin{bmatrix}B_1\\B_2\\B_3\end{bmatrix}=\begin{bmatrix}A_2B_3-A_3B_2\\A_3B_1-A_1B_3\\A_1B_2-A_2B_1\end{bmatrix}=(A_2B_3- A_3B_2)\hat{i} + (A_3B_1- A_1B_3)\hat{j} + (A_1B_2-A_2B_1)\hat{k}\)
   
   Similarly, the Matrix Product \([\vec{B}_{\mathbf{\times}}]\vec{A}\) for Cross Product \(\vec{B} \times \vec{A}\) is calculated as
   
   \(\vec{B} \times \vec{A}=[\vec{B}_{\mathbf{\times}}]\vec{A}=\begin{bmatrix}0 & -B_3 & B_2\\B_3 & 0 & -B_1\\-B_2 & B_1 & 0\end{bmatrix}\begin{bmatrix}A_1\\A_2\\A_3\end{bmatrix}=\begin{bmatrix}A_3B_2-A_2B_3\\A_1B_3-A_3B_1\\A_2B_1-A_1B_2\end{bmatrix}=(A_3B_2-A_2B_3)\hat{i} + (A_1B_3-A_3B_1)\hat{j} + (A_2B_1-A_1B_2)\hat{k}\)
   
   Also since \(\vec{A}_{\mathbf{\times}}\) and \(\vec{B}_{\mathbf{\times}}\) are Skew-Symmetric Matrices (i.e. their Negative is their Transpose), the following equations hold true
   
   \(\vec{A} \times \vec{B}= - \vec{B} \times \vec{A} = - [\vec{B}_{\mathbf{\times}}]\vec{A}= {[\vec{B}_{\mathbf{\times}}]}^T\vec{A}\)
   
   \(\vec{B} \times \vec{A}= - \vec{A} \times \vec{B} = - [\vec{A}_{\mathbf{\times}}]\vec{B}= {[\vec{A}_{\mathbf{\times}}]}^T\vec{B}\)
8. Following examples demonstrates the calculation of the Cross Product of Vectors \(\vec{A}\) and \(\vec{B}\) defined below using both Determinant Method and Matrix Product Method
   
   \(\vec{A}=-2\hat{i} + 5\hat{j} + 7\hat{k}\hspace{.5cm}\vec{B}=3\hat{i} - 6\hat{j} - 4\hat{k}\)
   
   Using Determinant Method the Cross Products are calculated as
   
   \(\vec{A} \times \vec{B} =\begin{vmatrix}\mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}}\\-2 & 5 & 7\\3 & -6 & -4\end{vmatrix}=(5\cdot(-4) - 7 \cdot (-6))\mathbf{\hat{i}} + (7 \cdot 3 - (-2) \cdot (-4))\mathbf{\hat{j}} + ((-2)\cdot(-6) - 5 \cdot 3)\mathbf{\hat{k}}=22\mathbf{\hat{i}} + 13\mathbf{\hat{j}} - 3\mathbf{\hat{k}}\)
   
   \(\vec{B} \times \vec{A} =\begin{vmatrix}\mathbf{\hat{i}} & \mathbf{\hat{j}} & \mathbf{\hat{k}}\\3 & -6 & -4\\-2 & 5 & 7\end{vmatrix}=(7 \cdot (-6) - 5\cdot(-4))\mathbf{\hat{i}} + ((-2) \cdot (-4) - 7 \cdot 3)\mathbf{\hat{j}} + ( 5 \cdot 3 - (-2)\cdot(-6))\mathbf{\hat{k}}=-22\mathbf{\hat{i}} - 13\mathbf{\hat{j}} + 3\mathbf{\hat{k}}\)
   
   Using Matrix Product Method the Cross Products are calculated as
   
   \(\vec{A} \times \vec{B} =[\vec{A}_{\mathbf{\times}}]\vec{B}=\begin{bmatrix}0 & -7 & 5\\7 & 0 & 2\\-5 & -2 & 0\end{bmatrix}\begin{bmatrix}3\\-6\\-4\end{bmatrix}=\begin{bmatrix}(-7) \cdot (-6) + 5 \cdot (-4)\\7 \cdot 3 + 2 \cdot (-4)\\(-5) \cdot 3 + (-2) \cdot (-6)\end{bmatrix}=\begin{bmatrix}22\\13\\-3\end{bmatrix}=22\mathbf{\hat{i}} + 13\mathbf{\hat{j}} - 3\mathbf{\hat{k}}\)
   
   \(\vec{B} \times \vec{A} =[\vec{B}_{\mathbf{\times}}]\vec{A} =\begin{bmatrix}0 & 4 & -6\\-4 & 0 & -3\\6 & 3 & 0\end{bmatrix}\begin{bmatrix}-2\\5\\7\end{bmatrix}=\begin{bmatrix}4 \cdot 5 + (-6)\cdot 7\\(-4) \cdot (-2) + (-3) \cdot 7 \\6 \cdot (-2) + 3 \cdot 5\end{bmatrix}=\begin{bmatrix}-22\\-13\\3\end{bmatrix}=-22\mathbf{\hat{i}} - 13\mathbf{\hat{j}} + 3\mathbf{\hat{k}}\)
9. When Cross Product is carried out between 2 Vectors, it makes the 2nd Vector Project on a Plane Perpendicular to the 1st Vector, and then Rotate the Projected Vector Counter-Clockwise by \(90^\circ\) by using 1st Vector as the Axis of Rotation (i.e the Rotated Projected Vector is the Resultant Vector of the Cross Product).
   
   For example, for Cross Product \(\vec{A} \times \vec{B}\), the Vector \(\vec{B}\) gets Projected on a Plane Perpendicular to \(\vec{A}\), and then the Projected Vector is Rotated Counter-Clockwise by \(90^\circ\) by using Vector \(\vec{A}\) as the Axis of Rotation (which becomes the Resultant Vector of the Cross Product \(\vec{A} \times \vec{B}\)).
   
   Similarly, for Cross Product \(\vec{B} \times \vec{A}\), the Vector \(\vec{A}\) gets Projected on a Plane Perpendicular to \(\vec{B}\), and then the Projected Vector is Rotated Counter-Clockwise by \(90^\circ\) by using Vector \(\vec{B}\) as the Axis of Rotation (which becomes the Resultant Vector of the Cross Product \(\vec{B} \times \vec{A}\)).
   
   This property of Cross Products helps in building Projection-Rotation Trasformation Matrices corresponding to Cross Products as follows
   
   Projection-Rotation Transformation Matrix for \(\vec{A} \times \vec{B} = BA^T-AB^T\)
   
   Projection-Rotation Transformation Matrix for \(\vec{B} \times \vec{A} = AB^T-BA^T\)
   
   Projection-Rotation Transformation Matrices are Skew Symmetric Matrices.
10. Multiplying any Vector \(\vec{C}\) with the Projection Rotation Transformation Matrix corresponding to \(\vec{A} \times \vec{B}\) (which is same as Calculating \((\vec{A} \times \vec{B}) \times \vec{C}\)) first Projects the Vector \(\vec{C}\) on a Plane Perpendicular to \(\vec{A} \times \vec{B}\) (which is same as the Plane containing the Vectors \(\vec{A}\) and \(\vec{B}\)) and then Rotates the Projected Vector Counter-Clockwise by \(90^\circ\) by using Vector \(\vec{A} \times \vec{B}\) as Axis of Rotation (in same Direction as Counter Clockwise Rotation from \(\vec{A}\) to \(\vec{B}\)).
    
    Similarly, Multiplying any Vector \(\vec{C}\) with the Projection Rotation Transformation Matrix corresponding to \(\vec{B} \times \vec{A}\) (which is same as Calculating \((\vec{B} \times \vec{A}) \times \vec{C}\)) first Projects the Vector \(\vec{C}\) on a Plane Perpendicular to \(\vec{B} \times \vec{A}\) (which is same as the Plane containing the Vectors \(\vec{A}\) and \(\vec{B}\)) and then Rotates the Projected Vector Counter-Clockwise by \(90^\circ\) by using Vector \(\vec{B} \times \vec{A}\) as Axis of Rotation (in same Direction as Counter Clockwise Rotation from \(\vec{B}\) to \(\vec{A}\)).
    
    If \(\vec{A}\) and \(\vec{B}\) are Mutually Orthogonal Unit Vectors then Projection-Rotation Transformation Matrices corresponding to \(\vec{A} \times \vec{B}\) and \(\vec{B} \times \vec{A}\) are Generator Matrices for Rotation Matrices for the Plane Containing Vectors \(\vec{A}\) and \(\vec{B}\).