To find the dot product of two vectors $\mathbf{a} = (2, -3, -7)$ and $\mathbf{b} = (1, -1, 2)$, you use the formula:

$\mathbf{a} \cdot \mathbf{b} = (a_1 \cdot b_1) + (a_2 \cdot b_2) + (a_3 \cdot b_3)$

Where:

• $a_1 = 2, a_2 = -3, a_3 = -7$
• $b_1 = 1, b_2 = -1, b_3 = 2$

Now, let's calculate:

$\mathbf{a} \cdot \mathbf{b} = (2 \cdot 1) + (-3 \cdot -1) + (-7 \cdot 2)$ $\mathbf{a} \cdot \mathbf{b} = 2 + 3 - 14$ $\mathbf{a} \cdot \mathbf{b} = -9$

So, the dot product $\mathbf{a} \cdot \mathbf{b} = -9$.

Let me know if you'd like any further details or have more questions!

Here are 5 related questions:

1. What is the geometric interpretation of the dot product?
2. How do you compute the angle between two vectors using the dot product?
3. What does it mean when the dot product of two vectors is zero?
4. How do you find the cross product of two vectors in 3D space?
5. Can you compute the magnitude of a vector? How is it done?

Tip: The dot product also measures how much one vector extends in the direction of another, and it is zero when the vectors are perpendicular.