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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.

Let \( \vec{a} = \langle 1, -3, -4 \rangle \) and \( \vec{b} = \langle -2, 0, 4 \rangle \). Find \( \vec{a} \cdot \vec{b} \).

Find the dot product of the vectors \( \vec{a} = \langle 1, -3, -4 \rangle \) and \( \vec{b} = \langle -2, 0, 4 \rangle \) with a step-by-step solution. This problem demonstrates the calculation of the dot product, a fundamental vector operation, suitable for high school students in Grades 10-12.

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