3Blue1Brown: Linear Algebra Chapter 10: Cross Products - Quiz

This quiz assesses deep understanding of the cross product, as introduced from both a standard and a more profound linear transformation perspective. It covers the geometric interpretation of the 2D cross product as a signed area and its computational link to the determinant. The quiz extends to the 3D cross product, focusing on its definition as a vector whose magnitude is the area of a parallelogram and whose direction is determined by the right-hand rule. Questions will challenge your knowledge of the underlying connections between determinants, linear transformations, and the geometric properties of vectors.

Key Formulae:

• 2D Cross Product (as a scalar):

  For vectors $V = \begin{pmatrix} v_1 \\ v_2 \end{pmatrix}$ and $W = \begin{pmatrix} w_1 \\ w_2 \end{pmatrix}$, the signed area is given by the determinant:

  $V \times W = \det \begin{pmatrix} v_1 & w_1 \\ v_2 & w_2 \end{pmatrix} = v_1 w_2 - v_2 w_1$

• 3D Cross Product (as a vector):

  For vectors $V = v_1\hat{i} + v_2\hat{j} + v_3\hat{k}$ and $W = w_1\hat{i} + w_2\hat{j} + w_3\hat{k}$, the cross product is computed via the symbolic determinant:

  $V \times W = \det \begin{pmatrix} \hat{i} & \hat{j} & \hat{k} \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{pmatrix}$