From: 3blue1brown
The cross product is an operation that has both a standard introduction and a deeper understanding related to linear transformations [00:18:00]. This article covers the typical geometric interpretations of the cross product in two and three dimensions [00:29:00].
Two-Dimensional Cross Product
In two dimensions, given two vectors v and w, their cross product, written as v × w, is directly related to the area of the parallelogram they span [01:02:00]. This parallelogram is formed by taking copies of v and w and moving their tails to the tips of the other vector [00:47:00].
Area and Orientation
The value of the 2D cross product is the area of this parallelogram, with an added consideration for orientation [01:06:00].
- If
vis to the right ofw,v × wis positive and equal to the parallelogram’s area [01:14:00]. - If
vis to the left ofw,v × wis negative, specifically the negative area of the parallelogram [01:21:00].
This implies that the order of the vectors matters; w × v will be the negative of v × w [01:28:00]. The orientation is defined by the order of basis vectors, such as i-hat × j-hat resulting in a positive value [01:48:00]. For example, if v is to the left of w and their parallelogram has an area of 7, then v × w is -7 [02:01:00].
Computation using Determinants
The 2D cross product can be computed using the determinant [02:20:00]. To find v × w, you create a matrix where the coordinates of v form the first column and the coordinates of w form the second column, then compute its determinant [02:37:00].
This method works because the determinant measures how areas change due to a linear transformation [03:06:00]. A matrix with v and w as columns represents a transformation that maps the unit square (defined by i-hat and j-hat) to the parallelogram spanned by v and w [02:51:00]. Since the unit square has an area of 1, the determinant gives the area of the transformed parallelogram [03:22:00]. A negative determinant indicates that the orientation was flipped during the transformation, which aligns with v being on the left of w [03:32:00].
For instance, if v = (-3, 1) and w = (2, 1), the determinant of the matrix [[-3, 2], [1, 1]] is (-3 * 1) - (2 * 1) = -5 [03:43:00]. This means the parallelogram has an area of 5, and the negative sign indicates v is to the left of w [04:01:00].
Intuitive Properties
- The cross product is larger when the vectors are closer to being perpendicular, as this creates a larger parallelogram area [04:19:00].
- Scaling one of the vectors scales the parallelogram’s area by the same factor. For example,
(3v) × wwill be three times the value ofv × w[04:37:00].
Three-Dimensional Cross Product
The “true” cross product combines two 3D vectors to produce a new 3D vector [05:05:00]. While the parallelogram spanned by the two vectors still plays a central role, the result is not a scalar area, but a vector [05:12:00].
Magnitude and Direction
- Magnitude: The length of the resulting 3D cross product vector is the area of the parallelogram formed by the two input vectors [05:30:00].
- Direction: The direction of this new vector is perpendicular to the plane containing the parallelogram [05:34:00].
Right Hand Rule
To determine the specific direction of the cross product vector, the right hand rule is used [05:50:00]:
- Point the forefinger of your right hand in the direction of the first vector (
v). - Stick out your middle finger in the direction of the second vector (
w). - Your thumb will then point in the direction of
v × w[05:53:00].
For example, if v points up in the z-direction (length 2) and w points in the y-direction (length 2), they form a square with an area of 4 [06:08:00]. Using the right hand rule, their cross product will point in the negative x-direction, resulting in a vector of negative 4 * i-hat [06:29:00].
Computation using 3D Determinant
A common method for computing the 3D cross product involves a specific process using a 3D determinant [06:45:00]. This involves setting up a 3D matrix where:
- The first column contains the basis vectors
i-hat,j-hat, andk-hat. - The second column contains the coordinates of
v. - The third column contains the coordinates of
w[06:55:00].
When the determinant of this matrix is computed by treating i-hat, j-hat, and k-hat as if they were numbers, the result is a linear combination of these basis vectors, which forms the cross product vector [07:25:00]. Students are often taught to accept this as a “notational trick,” but it’s crucial to understand that the resulting vector is indeed the unique vector that is perpendicular to v and w, has a magnitude equal to the parallelogram’s area, and its direction adheres to the right hand rule [07:40:00].
A deeper understanding of why the determinant is relevant here and the significance of placing basis vectors in the matrix involves the concept of duality, which is explored in a separate discussion [08:06:00]. The fundamental takeaway for geometric representation is the meaning of the cross product vector itself [08:23:00].