To find the area of the parallelogram formed by the vectors [tex]\(\mathbf{a} = 3\mathbf{i} - \mathbf{j} + 2\mathbf{k}\)[/tex] and [tex]\(\mathbf{b} = 4\mathbf{j} - 6\mathbf{k}\)[/tex], we will follow these steps:
1. Find the cross product of the vectors [tex]\(\mathbf{a}\)[/tex] and [tex]\(\mathbf{b}\)[/tex].
The cross product [tex]\(\mathbf{a} \times \mathbf{b}\)[/tex] is a vector that is perpendicular to both [tex]\(\mathbf{a}\)[/tex] and [tex]\(\mathbf{b}\)[/tex], and its magnitude represents the area of the parallelogram.
The components of the vectors are:
[tex]\[
\mathbf{a} = [3, -1, 2]
\][/tex]
[tex]\[
\mathbf{b} = [0, 4, -6]
\][/tex]
The cross product of two vectors [tex]\([a_1, a_2, a_3]\)[/tex] and [tex]\([b_1, b_2, b_3]\)[/tex] is given by:
[tex]\[
\mathbf{a} \times \mathbf{b} = \left( (a_2 b_3 - a_3 b_2) \mathbf{i} - (a_1 b_3 - a_3 b_1) \mathbf{j} + (a_1 b_2 - a_2 b_1) \mathbf{k} \right)
\][/tex]
Plugging in the components:
[tex]\[
\mathbf{a} \times \mathbf{b} = \left( (-1)(-6) - (2)(4) \right) \mathbf{i} - \left( (3)(-6) - (2)(0) \right) \mathbf{j} + \left( (3)(4) - (-1)(0) \right) \mathbf{k}
\][/tex]
Simplifying each component:
[tex]\[
\mathbf{a} \times \mathbf{b} = (6 - 8) \mathbf{i} - (-18 - 0) \mathbf{j} + (12 - 0) \mathbf{k}
\][/tex]
[tex]\[
\mathbf{a} \times \mathbf{b} = -2 \mathbf{i} + 18 \mathbf{j} + 12 \mathbf{k}
\][/tex]
Therefore, the cross product is:
[tex]\[
\mathbf{a} \times \mathbf{b} = [-2, 18, 12]
\][/tex]
2. Calculate the magnitude of the cross product vector [tex]\([-2, 18, 12]\)[/tex].
The magnitude of a vector [tex]\([x, y, z]\)[/tex] is given by:
[tex]\[
\|\mathbf{v}\| = \sqrt{x^2 + y^2 + z^2}
\][/tex]
Substituting the components:
[tex]\[
\|\mathbf{a} \times \mathbf{b}\| = \sqrt{(-2)^2 + 18^2 + 12^2}
\][/tex]
[tex]\[
\|\mathbf{a} \times \mathbf{b}\| = \sqrt{4 + 324 + 144}
\][/tex]
[tex]\[
\|\mathbf{a} \times \mathbf{b}\| = \sqrt{472}
\][/tex]
The area of the parallelogram is:
[tex]\[
\text{Area} = \sqrt{472}
\][/tex]
Comparing this result to the given options, the closest match is [tex]\(\sqrt{573}\)[/tex]. However, none of the options exactly matches [tex]\(\sqrt{472}\)[/tex], indicating there might be an error in the provided options. Assuming we should select the one closest to our result, the best match here is the answer closest to the computed area:
The correct answer is not listed among the options, but based on our computation, the area is:
[tex]\[
21.72556098240043 \, \text{units}
\][/tex]
