To determine if the vectors [tex]\(\overrightarrow{\vec{x}}+6 \vec{b}+7 \vec{c}\)[/tex], [tex]\(7 \vec{\lambda}-8 \vec{b}+9 \vec{c}\)[/tex], and [tex]\(3 \vec{x}+20 \vec{b}+5 \vec{c}\)[/tex] are coplanar, we need to ensure that their scalar triple product is zero.
The scalar triple product of three vectors is represented by the determinant of the matrix formed by their components. The vectors given are:
1. [tex]\(\overrightarrow{\vec{x}}+6 \vec{b}+7 \vec{c}\)[/tex]
2. [tex]\(7 \vec{\lambda}-8 \vec{b}+9 \vec{c}\)[/tex]
3. [tex]\(3 \vec{x}+20 \vec{b}+5 \vec{c}\)[/tex]
We represent these vectors as rows in a matrix:
[tex]\[
\begin{bmatrix}
1 & 6 & 7 \\
7 & -8 & 9 \\
3 & 20 & 5
\end{bmatrix}
\][/tex]
To determine if these vectors are coplanar, we calculate the determinant of this matrix and set it to zero. The determinant is given by:
[tex]\[
\begin{vmatrix}
1 & 6 & 7 \\
7 & -8 & 9 \\
3 & 20 & 5
\end{vmatrix}
\][/tex]
Calculate the determinant using the method of co-factors:
[tex]\[
\text{Det} = 1 \cdot \begin{vmatrix} -8 & 9 \\ 20 & 5 \end{vmatrix} - 6 \cdot \begin{vmatrix} 7 & 9 \\ 3 & 5 \end{vmatrix} + 7 \cdot \begin{vmatrix} 7 & -8 \\ 3 & 20
\end{vmatrix}
\][/tex]
Now, compute each of these 2x2 determinants:
1.
[tex]\[
\begin{vmatrix}
-8 & 9 \\
20 & 5
\end{vmatrix} = (-8)(5) - (9)(20) = -40 - 180 = -220
\][/tex]
2.
[tex]\[
\begin{vmatrix}
7 & 9 \\
3 & 5
\end{vmatrix} = (7)(5) - (9)(3) = 35 - 27 = 8
\][/tex]
3.
[tex]\[
\begin{vmatrix}
7 & -8 \\
3 & 20
\end{vmatrix} = (7)(20) - (-8)(3) = 140 + 24 = 164
\][/tex]
Now substitute these values back into the determinant expression:
[tex]\[
\text{Det} = 1(-220) - 6(8) + 7(164) = -220 - 48 + 1148 = 880
\][/tex]
This gives the determinant as 880. Since the scalar triple product (determinant) is not zero, the vectors are not coplanar, which contradicts the original statement about coplanarity.
Thus, there's no specific value of [tex]\( \vec{c} \)[/tex] that will make the vectors coplanar, as no solution (value of [tex]\(\vec{c}\)[/tex]) satisfies the given condition.
Therefore, the determinant is [tex]\(880\)[/tex], and there are no values for [tex]\( \vec{c} \)[/tex] that satisfy the given coplanarity condition.
