What is the determinant, cross product and dot product in short?

What is the Determinant?

The determinant is the area or volume of two or more vectors. The sum of the determinant tells us what happened after a transformation, namely if space has been flipped, squished into a smaller dimension, or scaled. The following formula calculates it in 2D space:

\[det\left(\begin{bmatrix} a & b \\\ c & d \end{bmatrix}\right) = ad-bc\]

What is the Cross Product?

The result of a cross product is a new vector with some length. The sum of the cross product is the length, and the vector points perpendicular to the plane of 2 vectors in 3D. The formula is as follows:

\[\begin{bmatrix}v_1 \\\ v_2 \\\ v_3\end{bmatrix}\times\begin{bmatrix}w_1 \\\ w_2 \\\ w_3\end{bmatrix} = det\left(\begin{bmatrix}\hat{i} & v_1 & w_1 \\\ hat{j} & v_2 & w_2 \\\hat{k} & v_3 & w_3\end{bmatrix}\right) = \\\ \hat{i}(v_2w_3-v_3w_2)+\hat{j}(v_3w_1-v_1w_3)+\hat{k}(v_1w_2-v_2w_1)\]

What is the Dot Product

The result of the dot product tells us where a vector is projected onto another vector, e.g., where it lands if we move it onto another vector. The sum of the dot product tells us (much like the determinant) in which way the projected vector is pointing, the same direction, opposite direction, or stuck at origin after projection. The formula for the dot product is as follows:

\[\begin{bmatrix}a \\\ b \\\ c\end{bmatrix} \boldsymbol{\cdot} \begin{bmatrix} d \\\ e \\\ f \end{bmatrix}=\begin{bmatrix}a \times d \\\ b \times e \\\ c\times f\end{bmatrix}=ad+be+cf\]