Section 5.3.3: Cross Product of Vectors and Matrix Determinants

Activity 5.10: Preliminary Questions on Vector Operations and Determinants

• Question 1: Foundational Knowledge   - The activity begins by prompting students to reflect on their existing knowledge regarding the cross product of vectors: "What do you know about cross product of vectors?"
• Question 2: Evaluation of a $2 \times 2$ Determinant   - Find $det(A)$ given the matrix: $A = \begin{pmatrix} 2 & 5 \ -1 & 3 \end{pmatrix}$
• Question 3: Evaluation of a $3 \times 3$ Determinant   - Find $det(A)$ given the matrix: $A = \begin{pmatrix} 4 & 5 & 1 \ 0 & -3 & 6 \ -2 & 1 & 4 \end{pmatrix}$

Determinant Calculation for $2 \times 2$ and $3 \times 3$ Matrices

• Definition of a $2 \times 2$ Determinant   - Recalling from the previous unit, the determinant of a $2 \times 2$ matrix is defined as follows:   - Given a matrix with components $u_1$, $u_2$ in the first row and $v_1$, $v_2$ in the second row, the determinant is represented as: $det \begin{pmatrix} u_1 & u_2 \ v_1 & v_2 \end{pmatrix} = u_1 v_2 - u_2 v_1$
• Expansion of a $3 \times 3$ Determinant   - A $3 \times 3$ determinant can be defined in terms of contributing $2 \times 2$ determinants through the following expansion rule:   - $det \begin{pmatrix} u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \ w_1 & w_2 & w_3 \end{pmatrix} = u_1 \begin{vmatrix} v_2 & v_3 \ w_2 & w_3 \end{vmatrix} - u_2 \begin{vmatrix} v_1 & v_3 \ w_1 & w_3 \end{vmatrix} + u_3 \begin{vmatrix} v_1 & v_2 \ w_1 & w_2 \end{vmatrix}$
• Structural Observations in Expansion   - Each term on the right side of the above equation involves a number from the first row of the determinant ($u_1$, $u_2$, or $u_3$).   - Any specific number $u_i$ is multiplied by a $2 \times 2$ determinant.   - This $2 \times 2$ determinant is obtained by deleting the specific row and column in which the multiplier $u_i$ appears.

Definition of the Standard Basis Vectors and Vector components

• Standard Basis Vectors   - The cross product is defined in conjunction with the standard basis vectors for three-dimensional space:     - $i = (1, 0, 0)$     - $j = (0, 1, 0)$     - $k = (0, 0, 1)$
• Vector Representation   - These basis vectors allow we to represent vectors $u$ and $v$ as linear combinations:     - $u = u_1 i + u_2 j + u_3 k$     - $v = v_1 i + v_2 j + v_3 k$

The Cross Product of Two Vectors

• Determinant Form of the Cross Product   - The cross product of vectors $u$ and $v$, denoted as $u \times v$, is defined using the following determinant structure, where the first row consists of the basis vectors $i$, $j$, and $k$:   - $u \times v = \begin{vmatrix} i & j & k \ u_1 & u_2 & u_3 \ v_1 & v_2 & v_3 \end{vmatrix}$
• Sub-Determinant Expansion of the Cross Product   - Expanding this determinant yields the following expression in terms of $2 \times 2$ determinants:   - $u \times v = \begin{vmatrix} u_2 & u_3 \ v_2 & v_3 \end{vmatrix} i - \begin{vmatrix} u_1 & u_3 \ v_1 & v_3 \end{vmatrix} j + \begin{vmatrix} u_1 & u_2 \ v_1 & v_2 \end{vmatrix} k$
• Component-Wise Formula   - Performing the arithmetic for the $2 \times 2$ determinants results in the final cross product formula:   - $u \times v = (u_2 v_3 - u_3 v_2) i - (u_1 v_3 - u_3 v_1) j + (u_1 v_2 - u_2 v_1) k$

Geometric Interpretation and Spatial Context

• Orthogonality   - The result of the cross product, the vector $u \times v$, is defined as a vector that is at right angles (orthogonal) to both original vectors $u$ and $v$.
• Dimensionality   - It is explicitly noted that the operation of the cross product happens exclusively within the three-dimensions (3D space).