Just a quick note on notation. I usually use (x, y, z) to denote components of vectors, and use angle brackets to denote inner product.

It looks so strange to see i-hat and j-hat with that dot

Cofactor expansion along first row to find the determinant which led to find the cross product

Ty so much ❤

You might not know this, but the cross product is actually known as a "wedge product" and produces a "bivector". This is because in 3 dimensions, all bivectors are pseudovectors, and when you work in 4 dimensions, the formula is different. Also, there is a "triple wedge product" that produces a "trivector" that can be used to form an analogous function to the cross product in 4 dimensions, where a pseudovector that is orthogonal to all 3 vectors is the output of said function. Using multivectors, this can be used to apply to differential forms: 0D VGA differential multivector: {1} 1D: {1, dx} 2D: {1, dr, dA} = {1, dx, dy, dydx} = {1, dr, rdθ, rdrdθ} 3D: {1, dr, dS, dV} = {1, dx, dy, dz, dydz, dzdx, dxdy, dzdydx} = {1, dr, rdθ, dz, drdz, rdrdθ, rdθdz, rdzdrdθ} = {1, dρ, ρdφ, ρ*sin(φ)dθ, ρdρdφ, ρ*sin(φ)dρdθ, ρ^2*sin(φ)dφdθ, ρ^2*sin(φ)dρdφdθ} Where the differential multivector shows the 0-form, the possibilities for the 1-form, the possibilities for the 2-form, all of the way to the final form. This can be used in proving Green's Theorem where ∮ F ∙ dr = ∯ curl F ∙ dS using vectors and bivectors (as curl F = ∇ ^ F), and the Divergence Theorem where ∯ F ∙ dS = ∭(div F)dV. Also, vector-valued functions parametrize lines, while bivector-valued functions parametrize surfaces with r(t) = . Using the 2D VGA, we can derive the 2D vector-valued function r(t) = , as well as the 2D bivector function f(x, y). Using the 3D VGA, the 3D vector function is r(t) = , the 3D bivector function is S(s, t) = r(s) ^ r(t) = as dS = dr ^ dr. Of course, we've got the 3D trivector function f(x, y, z). There can also be bivector fields, where F(x, y, z) = , as well as digradients, ∇² = ∇ ^ ∇. The digradient of a function is defined as the bivector using the partial derivatives of a function: For example, for an R^2 -> R function, the digradient is a scalar that shows the square slope of the 3D surface at a given point, which also shares the square slope as the tangent plane at that point, similar to how the gradient showing the slope of the tangent line to a 2D curve. For an R^3 -> R function, the digradient is a bivector equal to where f_x refers to the partial derivative of f respect to x. The generic 2-form differential of an R^2 -> R^2 function = ∇² ∙ dS, and you can tell if a bivector field is conservative using the double integral test. You can derive the 2-divergence and the 2-curl of a bivector field: div² F = ∇² ∙ F, and curl² F = ∇² ^² F. In a multivariable calculus class, the hardest thing on the Calculus III curriculum is Stokes' Theorem, which involves changing a line integral of a vector field into a surface integral of the curl of said vector field, and then finding the normal of the result, ending with using a form of the differential of surface area to compute the original line integral. However, using bivectors, it is possible to convert a 3D line integral directly into a volume integral. To make a 3-form differential from a 1-form differential, you need a 3x3 matrix with 3 components. You only have a gradient and a vector field. However, if you calculate the digradient of the function to form 2 gradients, you have the 3 components. You just plug them into the matrix and then calculate the determinant to get the coefficient of the integrand of the corresponding volume integral. For example, ∮(0, 1) ∙ dr along the path = ∭curl² F ∙ dV = ∭1.5dzdydx = 1.5. This can be generalized to any number of dimensions, such as computing line integrals in 4D using Stokes' Theorem. In physics, the most common concept of dot and wedge products is the concept of work vs. torque. Both involve a force vector and a distance vector, but work is a dot product of force and distance, and torque is a wedge product. This is because the second moment of inertia (also known as rotational inertia) is also a wedge product between the radius and the first moment of inertia, or I = r ^ M. This is because angular velocity requires complex numbers to represent (as angles are dimensionless quantities, the ratio between a distance and its magnitude, causing it to be meters per meter). We can derive higher moments of inertia and solid angular motion using the 2-wedge product, so that I² = A ^² I, and derive 3-torque, or solid angular force.

For manual computation without determinants help it can be done like usual multiplication with distributive property but there's some rules to be obeyed : 1) i x i = j x j = k x k = 0 2) i x j = k 3) i x k = -j 4) j x k = i 5) Reversing the order means you put a negative sign

sin3x=sin2x+¹

2+2=1 how?

No it's actually just this cross(u,v)