2.2. Cross Product

The cross product of two vectors yields a vector due to which it is also called vector product. In a coordinate system, say (e₁, e₂, e₃), the cross product can be written as follows:

(7)

The cross product is a mathematical operation that takes two vectors and produces a third vector. Unlike the dot product, which results in a scalar (a single number), the cross product results in a vector. This new vector is perpendicular to the plane formed by the two original vectors. The notation used in the lecture indicates that if you have vectors 'a' and 'b,' their cross product can be denoted as 'a × b.'

Think of the cross product like the way your hand would be oriented if you were holding two perpendicular sticks at their ends. Your fingers might point along one stick, while your palm points away from the other. The direction of your palm (representing the cross product) indicates the direction perpendicular to both sticks.

Thus, the cross product of two vectors can also be realized as the product of a skew-symmetric matrix (a matrix whose diagonal elements are 0 and off-diagonal elements are negative of each other) times the column of the second vector. The components of the skew-symmetric matrix are formed by the components of the first vector a. In order to easily remember how to form the skew-symmetric matrix from the components of a, one can remember the following trick: to get a component in the ith row and jth column, the component of a that will be used will be the third index (other than i and j). For example, for 1st row and 2nd column of the matrix, the third component a will be used.

A skew-symmetric matrix is a special kind of matrix that helps represent the cross product in a compact form. The matrix is said to be skew-symmetric because its elements satisfy the property that the element at position (i, j) is the negative of the element at position (j, i). This matrix can be used to compute the cross product without directly using the sine and angle but rather by matrix multiplication.

Imagine you have a 3D model of a rotating object. The skew-symmetric matrix acts like a rotation tool, allowing you to visualize how the product of two vectors results in a new vector that indicates the axis around which the object would rotate when you push it in one direction or another.

Geometrically, it is defined as follows:

a × b = ||a|| ||b|| sin(θ) c Here, c is a unit vector perpendicular to the plane formed by a and b.

This equation shows that the magnitude of the cross product is determined by the lengths of the two vectors and the sine of the angle between them. The angle θ is crucial as it determines how much the two vectors are 'leaning away' from each other. The direction of the resulting vector c is important too; it points outward from the plane created by the two vectors, following the right-hand rule.

You can visualize this while using a lever. If you push in two different directions on a lever, the force you apply creates a twist or rotation. The more angled your push is (maximized when it's perpendicular), the more 'twist' or rotational force you create. This twist is akin to the direction and magnitude of the cross product of the forces applied.

From equations of the dot product and cross product, we can observe that both dot product and cross product of two vectors are independent of the coordinate system. This is because the magnitude of vectors and the angle between the vectors do not change when we change the coordinate system.

This point underscores a fundamental aspect of vectors and their products: the physical principles represented by them don’t change regardless of how we choose to observe them. The coordinate system is simply a way to express the vector in a certain format but does not affect the actual properties of the vector itself, such as its direction or magnitude.

Imagine drawing a map. Whether you're using a traditional North-Up orientation or turning the map 90 degrees to face another direction, the actual landmarks and their relationships are unchanged; they just look different based on how you represent them on paper. Similarly, the vectors retain their inherent properties regardless of the coordinate system used.

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