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The scalar triple product is a valuable mathematical tool when analyzing the coplanarity of vectors. It gives us a scalar value by applying both the cross product and the dot product in succession. To determine whether three vectors are coplanar, we compute their scalar triple product and check if it equals zero.
This can be expressed as follows:

• Calculate the cross product of two vectors, creating a new vector.
• Apply the dot product between the resulting vector and the third vector.

In the format \[ \vec{u} \cdot (\vec{v} \times \vec{w}) \]if the result is zero, then the vectors are coplanar. The exercise involves vectors \( \vec{v_1} \), \( \vec{v_2} \), and \( \vec{v_3} \). Ensuring their scalar triple product equals zero confirms their coplanarity.

The cross product is crucial when dealing with vector operations that result in a product perpendicular to the initial vectors. It's particularly useful when finding the vector normal to a plane defined by two vectors—essential in determining coplanarity.
For vectors \( \vec{a} \) and \( \vec{b} \), the cross product \( \vec{a} \times \vec{b} \) creates a vector perpendicular to both, calculated using a 3x3 determinant:

• The determinant consists of unit vectors \( \vec{i}, \vec{j}, \vec{k} \).
• Values of the original vectors are inside the determinant.

In the given problem, the cross product of vectors \( \vec{v_2} \) and \( \vec{v_3} \) is solved using determinant expansion, which results in\[(yz + 4) \vec{i} + 2 \vec{j} - 2z \vec{k} \]. This vector will be then used for the dot product with \( \vec{v_1} \).

The dot product is another key vector operation that results in a scalar, representing the magnitude of one vector projected onto another. It is calculated by multiplying the corresponding components of two vectors and then summing these products. The formula is: \[ \vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3 \]In our exercise, once we have the result of the cross product, we compute the dot product with the third vector \( \vec{v_1} \). It's crucial because it simplifies to a scalar used to confirm the scalar triple product condition for coplanarity.
Given vectors are unit vectors with

• \( \vec{a} \cdot \vec{a} = 1 \)
• \( \vec{b} \cdot \vec{b} = 1 \)
• \( \vec{c} \cdot \vec{c} = 1 \)

This ensures the result simplifies cleanly to a meaningful outcome in terms of the variables \( x, y, z\).

Determinant expansion is a method for solving determinants in linear algebra, critical for vector operations like the cross product. The cross product of vectors can be computed by expanding a 3x3 matrix (determinant) using cofactor expansion.
The matrix is arranged as follows:

• Top row with unit vectors \( \vec{i}, \vec{j}, \vec{k} \)
• Subsequent rows represent the components of two vectors.

For the cross product \( \vec{v_2} \times \vec{v_3} \), the matrix is expanded as:\[\begin{vmatrix} \vec{i} & \vec{j} & \vec{k} -2 & y & -4 0 & -z & 2 \end{vmatrix} \]Cofactor expansion leads to terms multiplying \( \vec{i}, \vec{j}, \vec{k} \), providing crucial intermediate values necessary to compute other vector operations like the dot product. This step forms the backbone of verifying vector coplanarity in various contexts.