To find the dot product of the vectors $\mathbf{a} = (-1, -8, 7)$ and $\mathbf{b} = (1, 0, -10)$, use the formula for the dot product:

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

Substituting in the values:

$\mathbf{a} \cdot \mathbf{b} = (-1)(1) + (-8)(0) + (7)(-10)$

Calculating each term:

• First term: $(-1)(1) = -1$
• Second term: $(-8)(0) = 0$
• Third term: $(7)(-10) = -70$

Now, summing these results:

$\mathbf{a} \cdot \mathbf{b} = -1 + 0 - 70 = -71$

Thus, the dot product $\mathbf{a} \cdot \mathbf{b}$ is $-71$.

Would you like more details or have any questions? Here are some related questions:

1. What is the geometric interpretation of the dot product?
2. How would you calculate the dot product of two-dimensional vectors?
3. Can you explain how the dot product relates to the angle between two vectors?
4. What are the applications of the dot product in physics?
5. How do you determine if two vectors are orthogonal using the dot product?

Tip: Remember that the dot product can also indicate how parallel two vectors are; if it is zero, the vectors are orthogonal.