1. **State the problem:** Calculate the dot product $a \cdot b$ for the given vectors. 2. **Recall the dot product formula:** For vectors $a = \langle a_1, a_2 \rangle$ and $b = \langle b_1, b_2 \rangle$, the dot product is $$a \cdot b = a_1 b_1 + a_2 b_2$$ 3. **Apply the formula to the first pair:** Given $a = \langle 5, -2 \rangle$ and $b = \langle 3, 4 \rangle$, compute $$a \cdot b = 5 \times 3 + (-2) \times 4 = 15 - 8 = 7$$ 4. **Apply the formula to the second pair:** Given $a = \langle 1.5, 0.4 \rangle$ and $b = \langle -4, 6 \rangle$, compute $$a \cdot b = 1.5 \times (-4) + 0.4 \times 6 = -6 + 2.4 = -3.6$$ 5. **Apply the formula to the third pair (3D vectors):** For $a = \langle 6, -2, 3 \rangle$ and $b = \langle 2, 5, -1 \rangle$, the dot product is $$a \cdot b = 6 \times 2 + (-2) \times 5 + 3 \times (-1) = 12 - 10 - 3 = -1$$ **Final answers:** - For $a=\langle 5, -2 \rangle$, $b=\langle 3, 4 \rangle$, $a \cdot b = 7$ - For $a=\langle 1.5, 0.4 \rangle$, $b=\langle -4, 6 \rangle$, $a \cdot b = -3.6$ - For $a=\langle 6, -2, 3 \rangle$, $b=\langle 2, 5, -1 \rangle$, $a \cdot b = -1$