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Let's solve the given vector cross-product problem step by step. We are given two vectors:
- \(\vec{a} = \langle 3, -6, -7 \rangle\)
- \(\vec{b} = \langle 5, 2, 1 \rangle\)

We are asked to find:
\[
(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})
\]

### Step 1: Calculate \(\vec{a} - \vec{b}\)
\[
\vec{a} - \vec{b} = \langle 3, -6, -7 \rangle - \langle 5, 2, 1 \rangle = \langle 3 - 5, -6 - 2, -7 - 1 \rangle = \langle -2, -8, -8 \rangle
\]

### Step 2: Calculate \(\vec{a} + \vec{b}\)
\[
\vec{a} + \vec{b} = \langle 3, -6, -7 \rangle + \langle 5, 2, 1 \rangle = \langle 3 + 5, -6 + 2, -7 + 1 \rangle = \langle 8, -4, -6 \rangle
\]

### Step 3: Calculate the cross product \((\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})\)
The cross product of two vectors \(\langle x_1, y_1, z_1 \rangle\) and \(\langle x_2, y_2, z_2 \rangle\) is given by:
\[
\langle y_1 z_2 - z_1 y_2, z_1 x_2 - x_1 z_2, x_1 y_2 - y_1 x_2 \rangle
\]

Substituting the values of \(\vec{a} - \vec{b} = \langle -2, -8, -8 \rangle\) and \(\vec{a} + \vec{b} = \langle 8, -4, -6 \rangle\):

- First component:
\[
(-8)(-6) - (-8)(-4) = 48 - 32 = 16
\]

- Second component:
\[
(-8)(8) - (-2)(-6) = -64 - 12 = -76
\]

- Third component:
\[
(-2)(-4) - (-8)(8) = 8 - (-64) = 8 + 64 = 72
\]

Thus, the cross product is:
\[
(\vec{a} - \vec{b}) \times (\vec{a} + \vec{b}) = \langle 16, -76, 72 \rangle
\]

The final answer is:
\[
\boxed{\langle 16, -76, 72 \rangle}
\]

Would you like more details or have any further questions?

Here are some related questions to explore:
1. What is the magnitude of the cross product \(\langle 16, -76, 72 \rangle\)?
2. How would the result change if the vectors were in 4D space?
3. Can the cross product be extended to more than three dimensions?
4. What does the result of the cross product tell us geometrically about the two vectors?
5. How would you compute the dot product of \(\vec{a}\) and \(\vec{b}\) instead?

**Tip**: The cross product results in a vector perpendicular to both original vectors, useful in determining the normal to a plane defined by two vectors.

Let \(\vec{a} = \langle 3, -6, -7 \rangle\) and \(\vec{b} = \langle 5, 2, 1 \rangle\). Calculate \((\vec{a} - \vec{b}) \times (\vec{a} + \vec{b})\).

Solve the vector cross product problem involving \(\vec{a} = \langle 3, -6, -7 \rangle\) and \(\vec{b} = \langle 5, 2, 1 \rangle\). Learn how to calculate vector subtraction, addition, and the resulting cross product step by step.

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