To solve the given problem, we need to find the value of [tex]\( x \)[/tex] that satisfies the equation
[tex]$(3 \vec{i} - 2 \vec{j} + 2 \vec{k}) \cdot (2 \vec{i} - x \vec{j} + 3 \vec{k}) = -12.$[/tex]
This equation involves the dot product of two vectors. The dot product of two vectors [tex]\( \vec{a} = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k} \)[/tex] and [tex]\( \vec{b} = b_1 \vec{i} + b_2 \vec{j} + b_3 \vec{k} \)[/tex] is given by
[tex]\[ \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3. \][/tex]
Now, let's identify the components of our vectors:
- For the vector [tex]\( 3 \vec{i} - 2 \vec{j} + 2 \vec{k} \)[/tex]:
- [tex]\( a_1 = 3 \)[/tex]
- [tex]\( a_2 = -2 \)[/tex]
- [tex]\( a_3 = 2 \)[/tex]
- For the vector [tex]\( 2 \vec{i} - x \vec{j} + 3 \vec{k} \)[/tex]:
- [tex]\( b_1 = 2 \)[/tex]
- [tex]\( b_2 = -x \)[/tex]
- [tex]\( b_3 = 3 \)[/tex]
Substitute these components into the dot product formula:
[tex]\[ (3 \vec{i} - 2 \vec{j} + 2 \vec{k}) \cdot (2 \vec{i} - x \vec{j} + 3 \vec{k}) = 3 \cdot 2 + (-2) \cdot (-x) + 2 \cdot 3. \][/tex]
We can now simplify and solve for [tex]\( x \)[/tex]:
[tex]\[ 3 \cdot 2 + (-2) \cdot (-x) + 2 \cdot 3 = 6 + 2x + 6. \][/tex]
Combine like terms:
[tex]\[ 6 + 2x + 6 = 12 + 2x. \][/tex]
We set the dot product equal to [tex]\(-12\)[/tex] as given in the problem:
[tex]\[ 12 + 2x = -12. \][/tex]
To isolate [tex]\( x \)[/tex], we first subtract 12 from both sides:
[tex]\[ 2x = -12 - 12, \][/tex]
which simplifies to:
[tex]\[ 2x = -24. \][/tex]
Next, we divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{-24}{2} = -12. \][/tex]
This tells us that the value of [tex]\( x \)[/tex] is [tex]\(-12\)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{-12} \][/tex]
