To find the dot product $\vec{a} \cdot \vec{b}$, we use the formula:

$\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z$

where: $\vec{a} = 2 \hat{i} - 3 \hat{j} + 7 \hat{k}$ and $\vec{b} = -2 \hat{i} + 3 \hat{j} - 8 \hat{k}$

Step-by-Step Calculation:

1. Multiply the $\hat{i}$ components: $2 \cdot (-2) = -4$.
2. Multiply the $\hat{j}$ components: $-3 \cdot 3 = -9$.
3. Multiply the $\hat{k}$ components: $7 \cdot (-8) = -56$.

Now, add these results together:

$\vec{a} \cdot \vec{b} = -4 + (-9) + (-56) = -4 - 9 - 56 = -69$

Final Answer:

$\vec{a} \cdot \vec{b} = -69$

Would you like further explanation on vector dot products or any additional details?

Here are some related questions to expand on this topic:

1. What are the properties of the dot product?
2. How does the dot product relate to the angle between two vectors?
3. What would the result be if the vectors were orthogonal?
4. How does the dot product differ from the cross product?
5. Can the dot product be used to determine vector magnitudes?

Tip: The dot product of two vectors is zero if the vectors are perpendicular to each other.