To find the dot product $\mathbf{a} \cdot \mathbf{b}$ for the vectors $\mathbf{a}$ and $\mathbf{b}$, we use the following formula:

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

Given:

• $\mathbf{a} = \langle 7, -2 \rangle$
• $\mathbf{b} = \langle 3, 4 \rangle$

Substitute the components into the formula:

$\mathbf{a} \cdot \mathbf{b} = (7 \times 3) + (-2 \times 4)$

Calculate the products:

$\mathbf{a} \cdot \mathbf{b} = 21 + (-8) = 21 - 8 = 13$

So, the dot product $\mathbf{a} \cdot \mathbf{b} = 13$.

Would you like more details or have any questions?

Here are some related questions:

1. How can the dot product be used to find the angle between two vectors?
2. What is the physical interpretation of the dot product in physics?
3. How does the dot product relate to work done by a force?
4. Can the dot product be extended to vectors in three dimensions?
5. What are the properties of the dot product, such as distributivity and commutativity?

Tip: The dot product is a scalar quantity, meaning it has magnitude but no direction, and it's particularly useful for understanding projections and angles between vectors.