Given vectors:

$\mathbf{A} = 2\mathbf{i} - 3\mathbf{j} + \mathbf{k}$ $\mathbf{B} = \mathbf{j} - \mathbf{k}$

We are asked to find $(2\mathbf{A} - \mathbf{B}) \times (\mathbf{A} + \mathbf{B})$.

Step 1: Calculate $2\mathbf{A} - \mathbf{B}$

First, scale $\mathbf{A}$ by 2: $2\mathbf{A} = 2(2\mathbf{i} - 3\mathbf{j} + \mathbf{k}) = 4\mathbf{i} - 6\mathbf{j} + 2\mathbf{k}$ Now subtract $\mathbf{B}$: $2\mathbf{A} - \mathbf{B} = (4\mathbf{i} - 6\mathbf{j} + 2\mathbf{k}) - (\mathbf{j} - \mathbf{k}) = 4\mathbf{i} - 7\mathbf{j} + 3\mathbf{k}$

Step 2: Calculate $\mathbf{A} + \mathbf{B}$

Now add $\mathbf{A}$ and $\mathbf{B}$: $\mathbf{A} + \mathbf{B} = (2\mathbf{i} - 3\mathbf{j} + \mathbf{k}) + (\mathbf{j} - \mathbf{k}) = 2\mathbf{i} - 2\mathbf{j}$

Step 3: Cross Product $(2\mathbf{A} - \mathbf{B}) \times (\mathbf{A} + \mathbf{B})$

Now calculate the cross product between the vectors $(4\mathbf{i} - 7\mathbf{j} + 3\mathbf{k})$ and $(2\mathbf{i} - 2\mathbf{j})$.

We can compute the cross product using the determinant method: $\mathbf{v} \times \mathbf{w} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 4 & -7 & 3 \\ 2 & -2 & 0 \end{vmatrix}$

Expanding this determinant: $= \mathbf{i} \begin{vmatrix} -7 & 3 \\ -2 & 0 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 4 & 3 \\ 2 & 0 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 4 & -7 \\ 2 & -2 \end{vmatrix}$

Now calculate each 2x2 determinant:

$\begin{vmatrix} -7 & 3 \\ -2 & 0 \end{vmatrix} = (-7)(0) - (3)(-2) = 6$ $\begin{vmatrix} 4 & 3 \\ 2 & 0 \end{vmatrix} = (4)(0) - (3)(2) = -6$ $\begin{vmatrix} 4 & -7 \\ 2 & -2 \end{vmatrix} = (4)(-2) - (-7)(2) = -8 + 14 = 6$

Substitute these back into the cross product expression:

$= \mathbf{i}(6) - \mathbf{j}(-6) + \mathbf{k}(6)$ $= 6\mathbf{i} + 6\mathbf{j} + 6\mathbf{k}$

Final Result:

$(2\mathbf{A} - \mathbf{B}) \times (\mathbf{A} + \mathbf{B}) = 6\mathbf{i} + 6\mathbf{j} + 6\mathbf{k}$

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Would you like further explanation on any step or to explore related topics?

Here are some related questions:

1. How do you compute the cross product of two vectors geometrically?
2. How does the cross product relate to the area of a parallelogram?
3. Can you explain the distributive property of the cross product in more detail?
4. How do you handle cross products with unit vectors like $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$?
5. What happens if two vectors are parallel in a cross product?

Tip: The cross product is always perpendicular to the plane formed by the two original vectors.