To find the difference $\mathbf{v} - \mathbf{u}$ between the vectors $\mathbf{v} = \langle 7, -3 \rangle$ and $\mathbf{u} = \langle 3, 7 \rangle$, you subtract the corresponding components of $\mathbf{u}$ from $\mathbf{v}$:

$\mathbf{v} - \mathbf{u} = \langle 7, -3 \rangle - \langle 3, 7 \rangle$

Now, perform the subtraction:

$\mathbf{v} - \mathbf{u} = \langle 7 - 3, -3 - 7 \rangle$ $\mathbf{v} - \mathbf{u} = \langle 4, -10 \rangle$

So, the difference vector $\mathbf{v} - \mathbf{u}$ is $\langle 4, -10 \rangle$.

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 $\mathbf{u} = \langle 3, 7 \rangle$ and $\mathbf{v} = \langle 7, -3 \rangle$ 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 $\mathbf{v} - \mathbf{u} = \langle 4, -10 \rangle$ 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 $\mathbf{v} - \mathbf{u} = \langle 4, -10 \rangle$.

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

1. How do you compute the sum of two vectors?
2. What is the geometric significance of the vector $\mathbf{v} - \mathbf{u}$?
3. Can you calculate the magnitude of the vector $\mathbf{v} - \mathbf{u}$?
4. How can you find a vector perpendicular to $\mathbf{v} - \mathbf{u}$?
5. What is the dot product of vectors $\mathbf{u}$ and $\mathbf{v}$?

Tip: Drawing vectors on a coordinate plane helps visualize their sum, difference, and other operations!