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

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

where:

• $\vec{a} = \langle 1, -3, -4 \rangle$
• $\vec{b} = \langle -2, 0, 4 \rangle$

Now, substitute the values:

1. Multiply the first components: $1 \cdot (-2) = -2$
2. Multiply the second components: $-3 \cdot 0 = 0$
3. Multiply the third components: $-4 \cdot 4 = -16$

Adding these results together:

$\vec{a} \cdot \vec{b} = -2 + 0 - 16 = -18$

Thus, the answer is:

$\vec{a} \cdot \vec{b} = -18$

Would you like more details, or do you have any questions?

Here are some related questions for further practice:

1. What is the geometric interpretation of the dot product between two vectors?
2. If $\vec{a} \cdot \vec{b} = 0$, what can we say about the vectors $\vec{a}$ and $\vec{b}$?
3. How would you calculate the magnitude of vector $\vec{a}$?
4. What is the formula for the angle between two vectors using the dot product?
5. Can the dot product be used to determine if two vectors are parallel?

Tip: Remember that the dot product is a scalar quantity, not a vector, and gives information about the alignment of two vectors.