If \(\mathbf{A}=2 \mathbf{i}-\mathbf{j}-\mathbf{k}, \mathbf{B}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}, \mathbf{C}=\mathbf{j}+\mathbf{k},\) find \((\mathbf{A} \cdot \mathbf{B}) \mathbf{C}, \mathbf{A}(\mathbf{B} \cdot \mathbf{C}),(\mathbf{A} \times \mathbf{B}) \cdot \mathbf{C},\) \(\mathbf{A} \cdot(\mathbf{B} \times \mathbf{C}),(\mathbf{A} \times \mathbf{B}) \times \mathbf{C}, \mathbf{A} \times(\mathbf{B} \times \mathbf{C})\).

In vector algebra, the dot product (or scalar product) of two vectors measures how much one vector goes in the direction of another. The formula for the dot product of \(\textbf{A} = A_x \textbf{i} + A_y \textbf{j} + A_z \textbf{k}\) and \(\textbf{B} = B_x \textbf{i} + B_y \textbf{j} + B_z \textbf{k}\) is: \(\textbf{A} \cdot \textbf{B} = A_x B_x + A_y B_y + A_z B_z\). This calculation works by multiplying corresponding components of the vectors and summing the results. For example, given \(\textbf{A} = 2\textbf{i} - \textbf{j} - \textbf{k}\) and \(\textbf{B} = 2\textbf{i} - 3\textbf{j} + \textbf{k}\), the dot product is computed as: \((2)(2) + (-1)(-3) + (-1)(1) = 4 + 3 - 1 = 6\). The result is a scalar value.

The cross product (or vector product) of two vectors results in a vector orthogonal (perpendicular) to both original vectors. The formula involves the determinant of a matrix: The cross product of \( \textbf{A} \times \textbf{B}\) is given by: \(\textbf{A} \times \textbf{B} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \ \end{vmatrix}\). For our vectors, it means solving the determinant: \( \textbf{A} \times \textbf{B} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ 2 & -1 & -1 \ 2 & -3 & 1 \ \end{vmatrix} = \textbf{i}((-1)(1) - (-1)(-1)) - \textbf{j}((2)(1) - (-1)(2)) + \textbf{k}((2)(-3) - (2)(-1)) = -\textbf{i} - 6\textbf{j} - 4\textbf{k}\). The result is a new vector.

Vector algebra involves operations between vectors, including addition, subtraction, and multiplication (dot and cross products). Vectors can represent physical quantities such as force or velocity in physics. One can add vectors by summing their components: \( \textbf{A} + \textbf{B} = (A_x + B_x) \textbf{i} + (A_y + B_y) \textbf{j} + (A_z + B_z) \textbf{k}\). Subtraction follows a similar rule. Multiplication involves either the dot product, resulting in a scalar, or the cross product, which results in a vector. It is essential to understand these operations to solve many physics problems.

Vector multiplication includes both dot and cross products. The dot product measures parallelism, while the cross product measures perpendicularity. For two vectors \( \textbf{A}\) and \( \textbf{B}\), the dot product is computed as: \(\textbf{A} \cdot \textbf{B} = A_x B_x + A_y B_y + A_z B_z\). The cross product is presented as: \( \textbf{A} \times \textbf{B} = \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \ \end{vmatrix}\). Each product combines vector components in specific ways, either summing or forming new perpendicular vectors. Understanding these operations is crucial for analyzing physical phenomena.

Determinants are essential in vector calculations, especially for cross products. A determinant simplifies complex vector operations, providing a straightforward method for computing results. For example, in the cross product \(\textbf{A} \times \textbf{B}\), the determinant is expressed as a matrix involving the unit vectors and the components of \(\textbf{A}\) and \(\textbf{B}\). Solving: \(\begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \ A_x & A_y & A_z \ B_x & B_y & B_z \ \end{vmatrix}\), a methodical approach simplifies calculating essential outcomes in physics problems. By understanding and using determinants, vector multiplications become more manageable and visually interpretable.