91Ó°ÊÓ

The cross product, also known as the vector product, is a fundamental operation in vector algebra. It involves two vectors, producing a third vector that is perpendicular to the plane created by the original vectors. For vectors in three-dimensional space, the cross product of two vectors \( \boldsymbol{a} = (a_1, a_2, a_3) \) and \( \boldsymbol{b} = (b_1, b_2, b_3) \) can be expressed through a determinant:\[a \times b = \begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3\end{vmatrix}\]To compute this, one must calculate 3 sub-determinants, excluding one row and one column at a time for the components. Each sub-determinant provides a component for the resulting vector. This resulting vector is orthogonal, or perpendicular, to both \( \boldsymbol{a} \) and \( \boldsymbol{b} \). The direction follows the right-hand rule, meaning if you curl your right hand's fingers from \( \boldsymbol{a} \) to \( \boldsymbol{b} \), your thumb points in the direction of the cross product vector. Therefore, in our exercise: - \( a \times b = (0, 0, 1) \)- \( b \times c = (1, -1, 0) \)- \( c \times a = (0, 1, -1) \).These vectors lie perpendicular to the corresponding original vectors.

The dot product, also called the scalar product, is an operation that takes two vectors and returns a single number. This product provides information on the angle between the vectors as well as their lengths. Given vectors \( \boldsymbol{a} = (a_1, a_2, a_3) \) and \( \boldsymbol{b} = (b_1, b_2, b_3) \), the dot product is computed as:\[a \cdot b = a_1b_1 + a_2b_2 + a_3b_3\]The result is a scalar, hence the name scalar product. The dot product is particularly useful for determining orthogonality: if the dot product is zero, the vectors are perpendicular.In the problem, the dot product is used to compute the triple scalar product \( a \cdot (b \times c) \). This expression involves two operations: calculating the cross product \( b \times c \), and then taking the dot product of the result with vector \( \boldsymbol{a} \). The solution showed:- \( a \cdot (b \times c) = 1 \). This indicates that the three vectors \( \boldsymbol{a}, \boldsymbol{b} \), and \( \boldsymbol{c} \) have a geometrical volume in space related to the parallelepiped they form.

Linear combinations are a method in vector algebra of expressing a vector as a combination of others. This involves scaling vectors by real numbers, or coefficients, and adding them together. If you have vectors \( \boldsymbol{a} \), \( \boldsymbol{b} \), and \( \boldsymbol{c} \), and a vector \( \boldsymbol{d} \), expressing \( \boldsymbol{d} \) as a linear combination involves finding coefficients \( \alpha, \beta, \gamma \) such that:\[d = \alpha \boldsymbol{a} + \beta \boldsymbol{b} + \gamma \boldsymbol{c}\]The problem-solving approach is to set up a system of equations based on vector components, often leading to a straightforward algebraic solution. Here:- Using the components of \( \boldsymbol{d} = (2, -1, 2) \): - We found \( \gamma = 2 \) - \( \beta = -3 \) - \( \alpha = 3 \). This shows how \( \boldsymbol{d} \) can be "built" from the vectors given.

The triple scalar product involves three vectors \( \boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c} \) and is expressed as \( a \cdot (b \times c) \). It results in a scalar that is actually the volume of the parallelepiped formed by these vectors. It's computed as the dot product of one vector with the cross product of the other two. The properties of the triple scalar product make it both associative and distributive.In this specific exercise, the triple scalar product is crucial for validating expressions in terms of vector combinations:- \( a \cdot (b \times c) = 1 \), suggesting a unit volume in this geometrical context. - Solving for parameters \( p, q, r \) in cross product expressions involves this concept to express \( d \) in terms of the cross products \( a \times b, b \times c, c \times a \): - This use of the triple scalar product shows relationships between the vector and the vectors forming its basis, giving the needed coefficients once solved correctly, verifying the approach with both algebraic and geometric understanding.