To find the difference $\mathbf{v} - \mathbf{u}$ between the vectors $\mathbf{v} = \langle 4, -1 \rangle$ and $\mathbf{u} = \langle -5, -2 \rangle$, you subtract the corresponding components of $\mathbf{u}$ from $\mathbf{v}$:

$\mathbf{v} - \mathbf{u} = \langle 4, -1 \rangle - \langle -5, -2 \rangle$

Now, perform the subtraction:

$\mathbf{v} - \mathbf{u} = \langle 4 - (-5), -1 - (-2) \rangle$ $\mathbf{v} - \mathbf{u} = \langle 4 + 5, -1 + 2 \rangle$ $\mathbf{v} - \mathbf{u} = \langle 9, 1 \rangle$

So, the difference vector $\mathbf{v} - \mathbf{u}$ is $\langle 9, 1 \rangle$.

Graphical Representation

You can represent the difference vector $\mathbf{v} - \mathbf{u}$ graphically in two different ways:

1. Tip-to-Tail Method:

   • First, plot the vectors $\mathbf{u}$ and $\mathbf{v}$ starting from the origin.
   • Then, plot the vector $\mathbf{u}$ starting from the origin and the vector $\mathbf{v}$ starting from the tip of $\mathbf{u}$.
   • The vector $\mathbf{v} - \mathbf{u}$ is the vector that starts from the tail of $\mathbf{u}$ and ends at the tip of $\mathbf{v}$.

2. Parallelogram Method:

   • Plot both vectors $\mathbf{u}$ and $\mathbf{v}$ starting from the origin.
   • Draw a parallelogram using $\mathbf{u}$ and $\mathbf{v}$ as adjacent sides.
   • The diagonal of the parallelogram from the origin represents the vector $\mathbf{v} - \mathbf{u}$.

Would you like more details or have any questions about this solution? Here are some related questions:

1. How do you graphically represent the sum of two vectors?
2. What is the geometric interpretation of vector subtraction?
3. Can you find the magnitude of the difference vector $\mathbf{v} - \mathbf{u}$?
4. How do you find the unit vector in the direction of $\mathbf{v} - \mathbf{u}$?
5. What are the properties of vector addition and subtraction in general?

Tip: Always double-check the signs when subtracting vectors to avoid common mistakes!