1. **State the problem:** Compute the dot product of the vectors $\mathbf{a} = [14, 21, 28]$ and $\mathbf{b} = [4, 8, 20]$. 2. **Recall the formula:** The dot product of two vectors $\mathbf{a} = [a_1, a_2, a_3]$ and $\mathbf{b} = [b_1, b_2, b_3]$ is given by: $$\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3$$ 3. **Apply the formula:** Substitute the components: $$14 \times 4 + 21 \times 8 + 28 \times 20$$ 4. **Calculate each product:** $$14 \times 4 = 56$$ $$21 \times 8 = 168$$ $$28 \times 20 = 560$$ 5. **Sum the results:** $$56 + 168 + 560 = 784$$ 6. **Interpretation:** The dot product is a scalar value representing the sum of the products of corresponding components of the vectors. **Final answer:** $$\mathbf{a} \cdot \mathbf{b} = 784$$