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The dot product, also known as the scalar product, plays a crucial role in determining whether two vectors are perpendicular. When two vectors are perpendicular, their dot product equals zero.

To find the dot product of two vectors, say \( \boldsymbol{a} = (a_1, a_2, a_3) \) and \( \boldsymbol{b} = (b_1, b_2, b_3) \), you compute:\[ a_1b_1 + a_2b_2 + a_3b_3 \] The result is a scalar value.

For instance, let's examine vectors \( \boldsymbol{a} = (-1, -3, -1) \) and \( \boldsymbol{b} = (q, 1, 1) \). The dot product is \(-q - 3 - 1 = -q - 4\).
To determine when these vectors are perpendicular, we set the dot product equal to zero: \(-q - 4 = 0\), thus solving for \( q \) gives \( q = -4 \).
Therefore, when \( q \) is \(-4\), vectors \( \boldsymbol{a} \) and \( \boldsymbol{b} \) are perpendicular.

The cross product is a vector product that results in a vector perpendicular to two given vectors. It's a crucial operation in vector algebra, especially to find normals in 3D geometry.

To calculate the cross product of two vectors \( \boldsymbol{b} = (b_1, b_2, b_3) \) and \( \boldsymbol{c} = (c_1, c_2, c_3) \), we use the determinant as follows:

• Form a 3x3 matrix with the first row as the unit vectors \( \hat{i}, \hat{j}, \hat{k} \).
• Place vector \( \boldsymbol{b} \) in the second row and vector \( \boldsymbol{c} \) in the third row.

This results in:\[ \boldsymbol{b} \times \boldsymbol{c} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ b_1 & b_2 & b_3 \ c_1 & c_2 & c_3 \end{vmatrix} \]
Evaluating the determinant, we get each component of the resulting vector.

For example, with vectors \( \boldsymbol{b} = (q, 1, 1) \), and \( \boldsymbol{c} = (1, 1, q) \), the cross product is:\[(q-1)\hat{i} - (q^2-1)\hat{j} + (q-1)\hat{k}\].
The cross product vector \( \boldsymbol{b} \times \boldsymbol{c} \) becomes \( (q-1, -(q^2-1), q-1) \). This vector can be used to find further properties like coplanarity or parallelism.

The determinant is a key mathematical concept when computing the cross product using a matrix, especially in three dimensions. It’s vital in vector algebra for solving systems of equations and for identifying the characteristics of transformations.

When you compute the cross product, you set up a 3x3 matrix where the first row contains the unit vectors \( \hat{i}, \hat{j}, \hat{k} \), and the subsequent rows contain the components of the two vectors involved, \( \boldsymbol{b} \) and \( \boldsymbol{c} \).
The determinant of this matrix helps calculate each component of the resulting cross product vector.

For instance, in our scenario with vectors \( \boldsymbol{b} = (q, 1, 1) \) and \( \boldsymbol{c} = (1, 1, q) \), we have:\[ \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ q & 1 & 1 \ 1 & 1 & q \end{vmatrix} \]
To evaluate this determinant:

• For \( \hat{i} \): Compute \((1 \cdot q - 1 \cdot 1)\).
• For \( \hat{j} \): Compute \((-1)\times(q \cdot q - 1 \cdot 1)\).
• For \( \hat{k} \): Compute \((q \cdot 1 - 1 \cdot 1)\).

This results in the vector components \((q-1, -(q^2-1), q-1)\). The determinant allows us to find if certain conditions such as equilibrium or perpendicularity in vectors hold true.